Skip to main content

Full text of "Nonparametric comparison of slopes of regression lines"

See other formats


NONPARAMETRIC COMPARISON OF SLOPES 
OF REGRESSION LINES 



By 



RAYMOND RICHARD DALEY 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 
OF THE UNIVERSITY OF FLORIDA IN 
PARTIAL FULFILLMENT OF THE REQUIREMENTS 
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
1984 



■'•-.■= •;<r;->fl'-» 



To my parents 



ACKNOWLEDGEMENTS 

Dr. P.V. Rao suggested the research topic and, as my 
major professor, provided many hours of consultation with me 
during the preparation of this dissertation. For this I 
thank him sincerely. I also thank the other members of my 
committee, the Department of Statistics, and the staff of 
the Biostatistics Unit. I am grateful to Dr. Anthony Conti 
for allowing me flexibility in my work schedule that facil- 
itated completion of this dissertation. I will not forget 
Alice Martin for her practical advice and heartfelt support 
during the various qualifying exams. 

I thank my close friends for supporting my choice of 
academic goals v/hile showing me the benefits of a balanced 
life. I especially thank Christopher Kenward for his 
encouragement and positive attitude. Finally, I appreciate 
the love and support of my parents in all areas of m.y life 
and I am proud to share the attainment of this long-awaited 
goal with them. 



..4^_j»5>„— -fl 



TABLE OF CONTENTS 



Page 

ACKNOWLEDGEMENTS iii 

ABSTRACT V 

CHAPTER 

ONE INTRODUCTION 1 

1.1 The Basic Problem 1 

1 . 2 Literature Review 3 

1.3 Objectives and Overview 9 

TV70 COMPARING THE SLOPES OF TWO LINES 11 

2.1 Introduction 11 

2.2 Asymptotic Distributions 16 

2.3 Large Sample Inference 32 

2.4 Asymptotic Relative Efficiencies ... 35 

2.5 Small Sample Inference 68 

2.6 Monte Carlo Results 75 

THREE COMPARING THE SLOPES OF SEVERAL LINES ... 99 

3.1 Introduction 99 

3.2 Asymptotic Theory and a 

Proposed Test 104 

3.3 An Exact Test 138 

3.4 A Competitor to the Proposed Test . . . 141 

3.5 Asymptotic Relative Efficiencies . . . 161 

FOUR CONCLUSIONS 171 

BIBLIOGRAPHY 175 

BIOGRAPHICAL SKETCH 178 



vV 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 

NONPARAMETRIC COMPARISON OF SLOPES 
OF REGRESSION LINES 

By 

Raymond Richard Daley 

December, 19 84 



Chairman: Dr. Pejaver V. Rao 
Major Department: Statistics 



Distribution-free confidence intervals and point 
estimates are defined for the difference of the slope 
parameters in a linear regression setting with two lines, 
assuming common regression constants are used for both 
lines. A statistic, T, is proposed for this setting that is 
an application of a statistic discussed by Sievers in his 
paper titled "Weighted Rank Statistics for Simple Linear 
Regression," appearing on pages 628-631 of The Journal of 
the American Statistical Association in 1978. 

The statistic T employs weights, a , that are chosen 

by the user. If the regression constants are x^ , . . . ,x , 

then under the weights a = x -x , the null distribution of 

rs G r 

T depends on the regression constants. An. associated exact 
confidence interval for the slope difference can be obtained 
by calculation of a permutation distribution, requiring use 



of a computer. Under the weights a = 1 if x -x is 

^ rs s r 

positive, zero otherwise, the null distribution of T is 
essentially that of Kendall's tau. In this case, an associ- 
ated exact confidence interval for the slope difference can 
be calculated using readily available critical values of 
this distribution. 

Simulation results indicate the power of a test of 
parallelism based on T under either of the two sets of 
weights dominates the power of other available exact proce- 
dures. Pitman asymptotic relative efficiencies under equal 
spacing of the regression constants also favor a test based 
on T over other exact tests. Under unequal spacing of the 
regression constants, the simulation results suggest use of 

the weights a = x -x , when feasible. 
^ rs s r 

The method applied in the case of tv/o regression lines 
is generalized to construct a test for use in comparing the 
slopes of several regression lines with the slope of a 
standard line. The proposed test statistic is a quadratic 
form in a set of statistics, each having the structure of T. 
The asymptotic relative efficiency of the statistic with 
respect to a potential competitor is examined. 



vt' 



CHAPTER ONE 
INTRODUCTION 

o 

1.1 The Basic Problem 
There are many experimental settings where the appro- 
priate statistical analysis involves comparing the slopes of 
regression lines. For an example, consider a dilution assay 
in which two different drugs are being compared to get a 
measure of relative potency. A linear regression of the 
response on the chosen dosage levels is constructed for each 
drug. A fundamental assumption of this type of assay is the 
assumption that the two regression lines are parallel 
(Finney, 1964, p. 108). Hence a statistical test of paral- 
lelism of two regression lines would be desirable in this 
type of study. 

In other applications, an estimate of the difference in 
slopes might be needed. Consider a study comparing the 
effectiveness of two different fluoride dentifrices in 
reducing dental decay, where the participants engaged in 
varying numbers of supervised brushing sessions. The 
response measured is the change in the number of decayed and 
filled surfaces over the duration of the study, called the 
DFS increment. For each fluoride, the linear regression of 
DFS increment on number of supervised brushings usually 



results in a negative slope; as the number of supervised 
brushings increases, the average DFS increment tends to 
decrease. In this case the dental researcher might be 
interested in an estimate of the difference between the rate 
of decrease for the two fluoride groups, that is, an esti- 
mate of the difference of the slopes of the two regression 
lines. 

We have just offered two examples of the basic problem 
discussed in this dissertation, comparing the slopes of two 
or more regression lines. In this work we deal with linear 
regression models in the designed experiment, where the 
levels of the independent variable are determined by the 
experimenter, such as dose in a dilution assay. The methods 
of inference we propose assume the levels of the independent 
variable are chosen to be the same for each regression line, 
as is frequently the case in a designed experiment. Thus, 
in the dilution assay example, the chosen dosage levels for 
both drugs would be the same. 

Before proceeding further, we might mention that the 
problem of comparison of slopes does not suffer from lack of 
attention. As will be clear from the literature review in 
Section 1.2, there are many methods for comparing the slopes 
of two or more regression lines. The classical methods 
based on least squares theory are exact when the underlying 
distribution is normal and these methods are asymptotically 
distribution-free over the class of distributions having 
finite positive variance. Nonparametric methods exist which 



are either distribution-free or asymptotically distribution- 
free over a larger class of underlying error distributions, 
including heavily tailed ones such as the Cauchy distribu- 
tion under which the least squares methods perform poorly. 
The nonparametric, asymptotically distribution-free methods 
usually have good efficiency properties. Unfortunately, not 
only is their performance for small sample sizes suspect, 
but these methods are hard to implement in practice. In 
this work, we concentrate on the problem of developing 
better distribution-free methods of inference about slope 
differences in situations where the experimenter has control 
over the selection of the levels of the independent vari- 
able. 

1.2 Literature Review 
In this section we will review articles in the litera- 
ture discussing nonparametric analysis of linear regression 
models. We focus on works that are pertinent to the topic 
of this dissertation, comparing the slopes of two or more 
regression lines. Often a nonparametric procedure designed 
to make inference about the slope of a single line may be 
adapted to the case of several lines. In Chapter Two we 
will indicate a simple method of adapting a test designed 
for the single line setting to the two line setting, when 
the levels of the independent variable for both lines are 
chosen to be the same. For these reasons we include a 



discussion of the techniques applicable in the simple 
(single line) linear regression setting. 

To begin the discussion, consider the simple linear 
regression model, 

Yj = a + exj + Ej, (1.2.1) 

j = 1, .,., N. The y's are the observed responses, a and B 
are unknown parameters, the x's are the known regression 
constants (levels of the independent variable), and the E's 
are unobservable random errors. In this setting, nonpara- 
metric tests based on rank statistics have been examined by 
many authors (Hoeffding, 1951; Terry, 1952). Hajek (1962) 
discussed the general properties of tests based on the 
linear rank statistics. 



N 

V = E (X -x)<j,(R ), (1.2.2) 

j=l J 3 



where R^ is the rank of Y^ among Y^ , Y^ , ..., Y^^, 9 is a 

score function which transforms the ranks to scores in an 

N 
appropriate way, and x = Z x./N. Hajek discussed 

j=l ^ 

asymptotic properties of these tests which depend on the 
chosen score function. Good discussions of properties of 
the class of linear rank statistics (1.2.2) are found in 
Hajek and Sidak (1967) and Randies and Wolfe (1979) . 



Tests of the hypothesis B = based on the linear rank 
statistics (1.2.2) can be used to derive estimates of g in 
the model (1.2.1) by the application of a general technique 
developed by Hodges and Lehmann (1963) . Since its appear- 
ance in the literature, this technique has been applied to 
estimation problems in a wide variety of settings. Because 
Hodges-Lehmann estimates are used frequently in this disser- 
tation, we now give a brief example describing this estima- 
tion technique. 

Suppose we assiime the simple linear model (1.2.1) but 
with zero intercept, 

Yj = exj + E., (1.2.3) 

j = 1, ..., N, and we use the statistic (the numerator of 
the Pearson product moment correlation) 



N 

Q = Z (X -x)Y. (1.2.4) 

j=l J -* 



to test B = against the alternative 6 5^ . Suppose 
further that the errors E , j = 1, ..., n, are symmetrically 
distributed about zero. Then 



N 
Q(6) = Z (x.-x) (Y.-Bx.) 

j=l J 3 3 



n'n'»)f,>S«?-»« 



N 
= Z (x^-x)E^ (1.2.5) 



has median zero. Thus a desirable property for an estimate 
B of 6 is that Q(3) be as near as possible to the median of 
the distribution of Q(3), that is, as near as possible to 
zero. Then the Hodges-Lehmann estimate of B based on Q is 

the value of B such that Q(B) = inf |q(B) |. Of course this 

B 

example is in a special setting, but it does provide the 
basic motivation for a Hodges-Lehmann estimate. 

Adichie (1967) applied the technique of Hodges and 
Lehmann to derive estimates of the slope and intercept in 
the simple linear regression setting (1.2.1) using linear 
rank statistics of the form (1.2.2). Jureckova (1971) 
generalized these estimates to the multiple regression 
setting. An undesirable feature of the methods of Adichie 
and Jureckova is that calculation of the estimates and 
confidence intervals requires iterative procedures. 

Sen (1968) gave a complete development of an estimate 
of slope in the simple linear regression setting (1.2.1), 
that had earlier been suggested by Theil (1950) . The 
estimate is the median of the slopes 



Y - Y 

^ . x< x^ , (1.2.6) 



v -, 



X r s 



which is easy to calculate and intuitively appealing. The 
Theil-Sen estimate can be derived by applying the Hodges- 
Lehmann technique to the test statistic 



Z2 sgn(x -X )sgn(Y -Y ) , (1.2.7) 

r<s ^ ^ ^ ^ 



where sgn (u) = 1, 0, -1 as u is greater than, equal to, or 
less than zero. The statistic (1.2.7) is the numerator of a 
statistic known as Kendall's (1955) tau, a measure of 
association between the pairs (x., Y.). Since the null 
(3=0) distribution of Kendall's tau is distribution-free and 
has been tabulated, and the statistic (1.2.7) has an easily 
invertible form, exact small-sample confidence intervals for 
6 are readily constructed using the Hodges-Lehmann technique 
applied to the method of Theil and Sen; no iterative proce- 
dure is necessary. 

Scholz (1977) and Sievers (1978) extended the method of 
Theil and Sen by using a weighted Kendall's tau. The 
definition of a statistic equivalent to the one defined by 
Sievers will be given in Chapter Two. Sievers also showed 
how to construct confidence intervals and point estimates 
for the slope 3 . Note that although Scholz ' s work appeared 
prior to Sievers, the Scholz reference is only an abstract 
of an unpublished paper and hence we hereafter refer to this 
technique as the Sievers-Scholz approach. Further 
discussion of the Sievers-Scholz approach will be given in 
Chapter Two . 



Having considered nonparametric methods of inference 
about the slope in the simple linear regression setting, we 
now consider those methods applicable to the multiple 
regression setting. Recall the method of Jureckova (1971) , 
which was an extension of Adichie ' s (1967) results to the 
multiple regression case. A test of the parallelism of 
k >_ 2 lines using Jureckova 's method requires the computa- 
tion of k individual Hodges-Lehmann estimates, each calcu- 
lated by an iterative technique. Sen (1969) and Adichie 
(1974) specify tests of parallelism of k ^ 2 lines that 
require only a single Hodges-Lehmann estimate of the overall 
slope, arrived at iteratively. The Sen and Adichie method 
has good efficiency properties. However neither Sen and 
Adichie 's approach, nor the technique of Jureckova, provides 
exact confidence intervals for the individual slopes. 

Jaeckel (1972) suggested estimating regression param- 
eters by minimizing a chosen dispersion function of the 
residuals. He showed that for a certain class of dispersion 
functions his estimates are asymptotically equivalent to 
Jureckova' s, but easier to compute. However, Jaeckel 's 
estimates are also only asymptotically distribution-free and 
require iterative computations. 

Of the methods reviewed, only the Theil-Sen and 
Sievers-Scholz statistics, applicable in the simple linear 
regression setting, have readily invertible forms enabling 
easy computation of exact distribution-free confidence 
intervals for the slope parameter. In the two line setting. 



.^i-,* "jy >cm M vfj-m^i^ f ^- HM— wwFmp ' ■■ " i f m i *y^f\\»r'» -^ >f , »■ -^ «Tr* tv-='*r"pTr>J>*- ■ 



two exact procedures are found in the literature. The 
first, due to Hollander (1970) , is based on a statistic in 
the form of the Wilcoxon signed rank statistic. The second 
exact procedure, due to Rao and Gore (1981) , is applicable 
when the regression constants of the two lines are equally 
spaced. The statistic of this second procedure takes the 
form of a Wilcoxon-Mann-Whitney two-sample statistic. These 
two procedures are used to test the null hypothesis that the 
two regression lines are parallel. The null distributions 
of the two statistics are distribution-free, and exact 
distribution-free confidence interals for the difference in 
slopes can be readily computed by applying the Hodges- 
Lehmann technique. More discussion about the Hollander and 
the Rao-Gore methods will be given in Chapter Two. 

1.3 Objectives and Overview 
Having reviewed the literature regarding nonparametric 
comparisons of slopes of regression lines, we now state 
concisely the two objectives of this dissertation. The 
first objective is to develop efficient, exact nonparametric 
methods for comparing the slopes of two regression lines 
when the researcher has control over the choice of the 
levels of the independent variable. The methods we propose 
will enable construction of exact distribution-free confi- 
dence intervals for the difference between the two slopes. 
The exact techniques of Hollander (1970) and Rao and Gore 
(1981) , discussed briefly in the previous section, will be 



10 



direct competitors of the methods we suggest. A comparison 
of these three techniques in Chapter Two using their Pitman 
asymptotic relative efficiencies and a simulation study will 
establish the superiority of the new methods whenever they ' 
are appropriate. 

The second objective of this work is to generalize the 
new methods suggested in Chapter Two to the setting where 
the slopes of several regression lines are being compared. 
In Chapter Three we will extend these new methods to the 
multiple line case when the purpose is to compare the slope 
of one of k lines, considered a standard or control, to the 
slopes of the k-1 other lines. A comparison of our proposed 
test to a modification of Sen's (1969) test will show that 
our proposed test, in- addition to allowing an exact 
distribution- free test for small samples not available with 
Sen's approach, is almost as efficient as the modification 
of Sen's test when the sample size is large and the error 
distribution is not too heavily tailed. 



CHAPTER TWO 
COMPARING THE SLOPES OF TWO LINES 

2.1 Introduction 

In this chapter we examine the case of two regression 
lines, assuming the same regression constants are used for 
both lines. In this section we first establish the notation 
for the linear regression model and the statistic used in 
this chapter. We then motivate the form of the statistic 
and give some background concerning its development. A 
special characterization of the Hodges-Lehmann estimate 
associated with the statistic is given. We close this 
section with a brief look at the contents of the rest of 
Chapter Two. 

Consider the linear regression model 



Y. . = a. + 6 .X . + E. . , (2.1.1) 



i=l,2, j = l,...,N, and x- _< x^ £ . . . £ x . In this model a,, 
a , B-, and g^ ^^® unknown regression parameters, the x's 
are known regression constants, and the E's are mutually- 
independent, unobservable random errors. We are interested 
here in making inference about the slope difference. 



11 



12 



6-,_2 = B-, - &2' Since the regression constants are assumed 
the sanie for both lines, we suggest the use of the 
Sievers-Scholz approach introduced in Section 1.2 when 
discussing techniques appropriate in the simple linear 
regression setting, now applied to the differences Z . = Y, 



Ij 
considered by Sievers and Scholz has the representation 



- '^2i' J=lf''wN. A statistic equivalent to the one 



T*(b) = {1/N)i:z a^^sgn(Z^-Z^-bx^_^) , (2.1.2) 

r<s s r rs 



where x = x - x , sgn (u) = -1,0,1 as u is less than, 

^ o o X^ 

equal to, or greater than zero, and a > are arbitrary 

rs — ■' 

weights with a = if x = x . 
^ rs r s 

Note that under (2.1.1), the differences Z., j=l, ,N, 

follow the linear model 



^j = ^1-2 "■ ^l-2^j -^ El-2,j' (2.1.3) 



where a^_2 = a^ - a^, e^_2 = 6^ - 63' ^^^ E^_2^ . = E^. - 
E_., j=l,...,N. In computing the differences Z., j=l,...,N, 
we have reduced the two line case to the simple linear 
regression case, enabling us to apply the approach of 
Sievers and Scholz to the two line case. Of course the 
assumption of common regression constants is crucial to this 
reduction. 

Let us now motivate the form, of T* (b) by first 
discussing a special case of (2.1.2) due to Theil and Sen. 



13 



* * 

Let Z . (b) = Z. - bx. and consider the pairs (x., Z . (b) ) , 

j=l,...,N. Under H : g = b, the value of x. has no 

* 
effect on the value of Z . (b) = a^_„ + E, _ .. However under 

H, : Pi_2 > tjf a larger x. will tend to result in a 

* 
relatively larger observed Z . (b) = a,_„ + (3^ --b)x. + 

^1-2 i* '^^^^ ^ method of constructing a statistic for 

testing H- : &-,_2 = b is to employ some measure of associa- 

* 
tion between the x. and the Z . (b) , j=l,...,N. 

Theil and Sen used this approach, selecting Kendall's 

tau as the measure of association: 



II sgn(x )sgn(Z*(b)-Z*(b)) 
r<s rs s r 



[II sgn(x )ZZ sgn(Z^ (b)-Z^ fb))] ^ 
t<u ^" v<w ^ 

ZZ sgn(x )sgn{Z -Z -bx ) 
r<s s r rs 



* U k ' (2.1.4) 



* 

Here N is the number of positive differences x - x , 

s r' 

1 <_ r < s <_ N (keep in mind x^ _< x„ _< . . . _< x ) , 

N 
i^) = N(N-l)/2, and we assume no ties among the Z's. The 

* 

. and Z . 
: 3 



statistic U-^(b) is Kendall's tau between the x. and Z . (b) , 



j=l,...,N, for each fixed value of b. 

Let S = (Z -Z^)/x^^, 1 < r < s < N, denote the slope 

of the line connecting the observed differences Z and Z . 

r s 

Since Z^ - Z^ - bx_^ > if and only if S > b, we see that 
s r rs rs 

Uj^, (b) is a function of these slopes. The numerator of U (b) 
is equal to the difference between the number of slopes. 



14 



S^ , that exceed b and the number of these slopes which are 
less than b. 

Comparing U (b) given by (2.1.4) with T (b) given by 
(2.1.2), we see the Sievers-Scholz statistic is an extension 
of the statistic due to Theil and Sen obtained by replacing 
sgn(x^ ) with a general weighting function a . This allows 
the slopes determined from points which are farther apart to 
be given more weight than the slopes determined from closer 
points. Under fairly general regularity conditions, Sievers 
has shown the highest efficacy of a test based on his 
statistic is attained when the slopes S are given weights 
a = X - X = X . Rewriting T (b) using the optimal 
weights, a = x , we define 

J. o XT S 



T(b) = (1/N)ZZ x^^sgn(Z^-Z^-bx^^) . (2.1.5) 

r<s s r rs 



In the remainder of this chapter, we explore the appropriate- 
ness of T(b) for inference about 6._2- 

We have seen the Sievers-Scholz statistic is a 
generalization of the statistic due to Theil and Sen. There 
is a similar relationship between Hodges-Lehmann estimates 
of the slope parameter associated with the Theil-Sen and 
the Sievers-Scholz statistics. A Hodges-Lehmann estimate of 
^1-2 ^^^^"^ °^ ^^^ Theil-Sen statistic can be shown to be 
equal to the median of the set of slopes 

{S^^: l_<r<s^N, X ^ x } (Sen, 1968) . The corres- 
ponding Hodges-Lehmann estimate associated with the 



15 



Sievers-Scholz statistic T(b) is a generalization of the 
Theil-Sen estimate and can be viewed {Sievers, 1978) as a 
median of the distribution of a random variable V, where 



P{V=v} = x^g/x.. if v=S^g, (2.1.6) 



X.. = EZ X , l_<t<u_<N, Xt^x. Thus the Sievers- 
t<u t: u 

Scholz estimate is a weighted median of the slope estimates 

{S^^: l<r<s<N, X t^x}. 
rs — — r s 

In the next section we briefly summarize Sievers' 
results concerning the asymptotic distribution of the 
Sievers-Scholz test statistic and estimate, now applied to 
the two regression line setting. Using these results, we 
describe large sample inference of e._2 in Section 2.3. 
Pitman asymptotic relative efficiencies (PAREs) of the 
Sievers-Scholz procedure with respect to other nonparametric 
approaches as well as the least squares procedure are given 
in Section 2.4. These PAREs are derived assuming equally 
spaced regression constants. In Section 2.5 we propose 
two exact tests of H^ : Q^_^ = which are easily implemented 
for small sample sizes. Finally, we close Chapter Two with 
a Monte Carlo study in Section 2.6. The first part of this 
study concentrates on comparisons of the Sievers-Scholz 
asymptotic procedure with others under moderately large 
samples, while the second deals with exact tests and small 
samples. Because PARE ' s are available only when the x's are 
equally spaced, these Monte Carlo simulations emphasize 



16 



comparisons of the various test procedures under unequally- 
spaced regression constants. 

2.2 Asymptotic Distributions 
In this section we present some important results 
concerning the asymptotic distributions of the Sievers- 
Scholz statistic and estimate. All of these results follow 
from straightforward modifications of Sievers' (1978) 
results for the simple linear regression setting. Sievers 
presents his theorems without proofs, giving reference 
primarily to the text by Hajek and Sidak (1967) . We repeat 
these results, now applying them to the two line setting. 
We indicate references to proofs or supply the basic steps, 
since some of the results in this section will be needed in 
Chapter Three when considering several regression lines. 

As mentioned in the previous section, we are assuming 
the optimal weights, a = x^ - x^, r < s, and thus we state 
the results for T(b) in (2.1.5). Note that under these 
optimal weights, T(b) can be expressed in terms of the ranks 
of the differences Z . (b) , j=l,...,N, as follows: 



N 

* 



T(b) = (2/N) Z [Rank(Z. (b))x.] - (N+l)5, (2.2.1) 

j=l ^ 3 



where Rank (Z . (b) ) is the rank of Z . (b) among {Z*(b): 

N 
r = 1,...,N} and x = Z x./N. Hence, T(b) is a linear rank 

j = l -• 



17 



statistic and distributional theory for this class of 
statistics applies to T (b) . For example, assuming some 
regularity conditions^ asymptotic normality of T(0) under 
Hq: e-]^_2 = is immediate (see Theorem 2.2.1). 

Basic notation . We now give some notation that will be 
maintained throughout Chapters Two and Three. Let 



2 N 
a = Z (x_.-x)^/N, 
^ j = l ^ 



(2.2.2) 



^T,A = ^(^j-^)^/3 = Na2/3, 



(2.2.3) 



s-1 

s j^;^ s : 



(2.2.4) 



X 



N 

Z 

j=r+l 



(x.-x ) , and so 



(2.2.5) 



x.. - X . . = N(x.-x) and 



(2.2.6) 



2 2^-2 

ZZ X = EZ (X -X )^ = N Z (x.-x)^, 

r<s ^^ r<s ^ ^ j=l ^ 



(2.2.7) 



by simple algebra. For ease in notation, we assume the 
underlying error distributions of the two lines are the 
same, with cumulative distribution function (cdf) F and 
probability density function (pdf) f. Let E^ , j==l,2,... 



■ff U ' HJjJl 'ii ii * w ^ . I 1 111 i »jn i j^j ii un.; ! u t[iiiy*wiwrywwiiig?WM'i"*'^a<>EWI Gil l JI, *M U p 



18 



designate independent and identically distributed random 
variables with cdf F and pdf f; G and g will denote, 
respectively, the cdf and pdf of E^ - E^ . The cdf and pdf 
°^ ■^l ~ ^2 ~ -^3 ^ ^4 ^^11 ^^ denoted by H and h, 
respectively. For general pdf g, let I (q) = fq^'{x)dx, 
provided the integral is finite. The normal distribution 
with mean y and variance a^ will be designated 
by the notation N{y,a ). We adopt the 0(O, o{'), 
°p^'^' °p^*^' ^^'^ '^ notations as in Serfling (1980, p. 1, 
8-9) . Let P indicate convergence in probabilitv and ^ 
indicate convergence in distribution. Finally, let a(b+c) 
be notation for the interval ( | ab | - | ac | , | ab | + | ac | ) . 

Conditions. The following conditions are used in the 
statements of the theorems of this section: 



N 

_ 2 2 

2 (x.-x) = Na -)■ 00 as N ->■ » 

j = l ^ , ^ CIO iN . 



II. max (x.-x) max (x.-x)^ 

l^J^N -^ l<;i^N - 

N 2 ^ ° as N ■> 

Z (X -5)2 Na^ 

j = l ^ 



III. G is continuous. 

IV. G has a square integrable absolutely continuous 
density g. 

Condition II, the familiar Noether (1949) condition for 
regression constants, ensures that no individual regression 
constant dominates the others in size as N ^ - 



19 



Theorem 2.2.1 ; Under H^ : ^^_^ = b and condition III, 
the exact variance of T (b) is [ (1/3 ) + (1/N) ] Na,^ . If 
^0" ^1-2 " ^ ^^'^ conditions 1, li, and III hold, then 



^^^)/''t,A - N^°'l) (2.2.8) 



as N ->■ «. 



Proof; 



From the definition of the Z.'s (2.1.3), 



's - ^ - ^^rs = ^1-2, s - ^1-2, r + <^l-2-^)^rs- <2.2.9) 



Clearly then, the distribution of T (b) (2.1.5) when 6, = b 
is the same as the distribution of T(0) when 6-, - = 0. So 
assuming ^-^_2 = 0, we derive the variance of T(0). 



Var[T(0)] = Var[(l/N) II x sgn (E, , - E, ^ ) 1 

r<s rs ^ 1-2, s 1-2, r'^ 

* ' r't's''^^"*^ =°''[=9°<=l-2,s-^l-2,r'' =5n (Ei.2,s-=l-2 ,t' ^ 

" ' r's'u''-^"^" '^°"'=9'"^l-2,s-^l-2,r'' =9" '=1-2 ,u-=l-2 , = ' " 
= (l/N^) {Cj + Cj + C3 + c^}, (2.2.10) 



f 



¥ * -- ' i--X' ^^g"g^g^|^gg^*W r^i5> "^; :jP!*Q»gJ g '«'**^M M P« *-L» B;i'' J w .i;» i Nai M« Ct . 



20 



where c^, c^, c^ , and c^ denote the 4 terms in this variance 
expression. Since E ^ ^ - E is symmetric about zero, 
it follows that E[sgn(E^_2 ^ - E^_^ ^) ] - o. Also since 



'2_2 s ~ ^1-2 r ^^ continuous, it follows that 

2 
sgn (E.|^_2 g - E^_2 j.) = 1 with probability one. Thus 



^1 = 11^ x's^^-f^^^ (^l-2,s-^l-2,r)^ 



11^ 4s ^f^^^' (^l-2,s-^l-2,r)^ 



22 X (2.2.11) 

r<s ^^ 



To simplify C2 we first note 

Gov [sgn (E^.s , g-E^.s ,^) , sgn (Ei_2 ^^-Ei_2 ^^) ] 

= E[sgn (Ei_2,3-Ei_2,r) sgn (Ei_2 ,^-Ei_2 ,P ] 
= t^ ^^1-2, s - ^1-2, r ' 0' ^l-2,u " ^1-2, r > '^ 
+ ^ ^^1-2, s - ^1-2, r < 0' ^1-2, u " ^1-2, r < 0>^ 

- f^ ^^1-2, s - ^1-2, r ^ °' El-2,u - ^1-2, r ^ 0> 

^ ^ ^V2,s - ^1-2, r ^ °' ^l-2,u - ^1-2, r > °>^ 
= [ /[l-F(e)]^dF(e) + /F^(e)dF(e)] 

- [ /[l-F(e) ]F(e)dF(e) + /F (e) [1-F (e) ] dF (e) ]■ 
= [(1/3) + (1/3)] - [2(1/6)] 



1-2 s "^^ continuous (condition III) and consequently ; 



21 



- l/3f (2.2.12) 



and so it follows that 



'2 ^ \<s</rs-ru ^^^^ ^^^ ^^1-2 ,s-^l-2 ,r) '^^^ ^^1-2 ,u-^l-2 ,r) ^ 



(1/3) (2 ZZS X X ) 

rs ru 
r<s<u 

(1/3) (I xj ) (using (2.2.5)). (2.2.13) 

r ^• 



Similar basic manipulations yield the following 
simplifications of c- and c . : 



'3 = (1/3) (Z X h, (2.2.14) 

r 



c^ = - (1/3) (2 Z x^^x^^). (2.2.15) 



Substituting these simplifications into (2.2.10) we have 

Var [T(0)] = (l/N^) { ZZ x^ 

r<s ^^ 

+ (1/3) Z (x^^ + x^^ - 2x^_x^^)} 

= (l/N^) {J:z xJ + (1/3) I (x^ -x )^}. (2.2.1&) 
r<s ^^ r ^' '^ 

Using (2.2.6) and (2.2.7) in (2.2.16) yields 



22 



Var[T(0)] = (1/N^){N E (x.-x)"^ + (1/3)N^ I {x.-S)^} 

j=l 3 j=i 3 



^ - 9 

= Z (x -x)^ { (1/N) + (1/3) } 
j = l J 

= [(1/3) + (1/N)] Na^, (2.2.17) 

o 

verifying the expression for the exact variance of T (b) 
under H^ : B = b. 

If we take k=2 in Theorem 3.2.3 then this theorem 
states that under H^ : e^_2 = 0' assuming conditions I, II, 
and III 



T(0)/a^^^ J N(0,1) (2.2.18) 



as N ^ 00. From the remarks at the beginning of this proof 
T(b)/a^^^ has the same limiting distribution under H_ : 
^1-2 ^ ^' ^^^ ^° ^^^ proof is complete. Note that 0^ is 
the asymptotic variance of T(b) under H : 3 ^ = b. 

Theorem 2.2.2 ; Assume a sequence of alternatives 

J, 
^N* ^1-2 "^ ^1-2^^^ "^ to/(N^a^) to the null hypothesis H- : 

2l_2 = 0' where to is a constant. Then under conditions 
I-IV, 



'^^°^/''t,A t N(2(3'S)a)I(g),l) (2.2.19) 



as N ->■ 00 



-•»w ajj< > u*j. i i i' m i L i' J»i » iii i i.w i-ii P , III 1 . 1 m m rajnu ,« i M« 



23 



Proof ; 

A detailed argument is given, since results obtained 
here are used in the proof of Theorem 3.2.4 in Chap- 
ter Three. 

Let <^i^ = 0)/ (N^o^) . From (2.2.9) and the definition of 
T(b) (2.1.5), it is clear that the distribution of T(0) 
under e^_2 = im^ is the sam.e as the distribution of T(-a) ) 
under ^-^^2 ~ ^' Therefore the proof is complete if we 
assume &2_-2 ^ ° ^^^ show that T (-cu ) has the desired 
limiting distribution. V7e first state and prove two lemmas. 



Lemma 2.2.3 : Assume 3^_ = and conditions I-IV. 



Then 



E[T (-03^)70^^^] ^ 2(3')a)I(g) (2.2.20) 



as N -»■ «>. 
Proof of lemma: 



E[T(-a3^)] = (1/N)ZE x^^E [ sgn (Z^-Z^+co^^x^^^) ] . 

r<s 



Now 



E[sgn(Zg-Z^+aj^x^g)] = 1 - 2/G (z-u^x^^) dG (z) . (2.2.21) 



^>^>»i^>*v. ->r»V ^>V'>_:.i-^r» 



24 



Conditions III and IV allow us to write a Taylor's series 

expansion of G(2-MX) : G(2-cux^) =G(z)-cax a(z-e (z)), 

j-NXo Mrs Nrs" rs 

where < |9^^(z)| £ ^N^^rs ' " ^^^""^ 



max X . -X 



^N^rsl = U(x^-V/(n\H < 2 M i£J|N_i 



N^ a 



X 



it follows from condition II that 6 (z) ^ uniformly in r 

rs -' 

and s as N -^ ". That is, for all e > 0, there exists N(e) 
such that 

N > N(e) => Icoj^x^gl < e for all r < s. (2.2.22) 

Then by absolute continuity of g (condition IV) , for all 
6 > 0, there exists N {&) such that 

* • 
N > N (5) => g(2-6^g(z)) e (g(2)±<S), (2.2.23) 

for all 2. Since g(-) and 6 are nonnegative and g(«) is 
square integrable (condition IV), multiplying by g(z) and 
integrating we see 



* 
N > N (6) => /g(z-e^g(z))g(z)dz e (I(g)±6). (2.2.24) 



Substituting the Taylor's series expansion into (2.2.21) we 

have 

E[sgn(Z^-Z^+o;j^x^^)] = 2a)^^x^^/g (z-e^^ (z) ) g (z) dz , 



-.rr- T^v-".;^!,!,. 



25 



and so 



E[T(-a.^)/a^^^] = (2a)^/ (Na^ ^^) ) Z Z x^^/g (z-G^^ (z ) ) g (z ) dz 

X ^ o 



Assume tu > 0. Using (2.2.24) in the above representation we 
see N > N (6) => 



E[T(-aj^)/a^ ^] e {2a)^/ (Na^ ^))IZ x^ (1(g) ±5) 
' ' r<s 



^ - 2 
(2a)/(NG ))(N Z (X -x)^) (I(g)±6) (from (2.2.7)) 

' -1=1 J 



2(3^)a)(I(g)±6) , 



and so, N > N* (6/ (2 (3 ') u) ) => 



E[T(-cj^)/a^^^]e(2(3^)ujI(g)±6) , 



and (2.2.20) is proved for w > . The proof is similar for 
for 0) < 0. 



Lemma 2.2.4 : Assume 6,_2 = and conditions I, II, and 
III. Then 






III f^\-*'\^ fm »-»^%r^J'imm'**t r'i:' mm mj m ft .jr 



26 



as N ^ " . 
Proof of lemma : 

Since under &^_^ = 0, E[T(-a)j^) - T(0)] - E[T(-aj^)], 
we prove (2.2.25) by showing 

Var[(T(-a)jj)-T(0))/a^^^] -^ (2.2.26) 

as N ^ ». First, T(-a)j^) - T(0) = ^IZ ^rs^rs ^"n^ ' 

where E^^it,^^) = sgn (Z^-Z^+oj^x^^) - sgn(Z^-Z^). 
Assuming co > , 



2 -tj„x < Z - Z < 
otherwise. 



We will suppress the argument ui^ of H^^ (u^^) . Then, since 
'Vrs' "^ ° uniformly in r and s (see (2.2.22)) and the cdf 

of 2g-Z^ is continuous (condition III) , it follows that 

2 

E[H^^] = 4P{-aj^x^^<Zg-Z^<0} ^ uniformly in r and s as 

N > ". Similarly, E[H H , ,] > uniformly in r, s, r' , 

s' as N > <». So Var(H ), Cov(H ,H ), and other terms of 

J- o r s r u 

that form are all uniformly o(l) as N ■» ». Now 



Var[(T(-a)^)-T(0))/a^^^] 



(N^aJ ) ^Var[EZ x H ] 
-' r<s ^^ ^^ 



lj*t*;:«|«e^»|,-i-«[j> 



27 



= (N2aJ^^)-l(ZE^x2^Var H^^ 



+ 2 ZEE X X Cov(H ,H ) 
rs ru rs ru 
r<s<u 



+ 2 ZEE x^^x, Cov(H ,H^ ) 
r<t<s ^^ ^^ ^^ t^ 



+ 2 EEE X X Cov(H ,H ) } 
r<s<u ^^ =^ ^^ ^^ 



2 2 -1 
" ^^ '^T,A^ (0^+02+03+0^), (2.2.27) 



where c^, <z^, o^, and c^ denote the 4 terms in this varianoe 

expression. Since r < s < u implies < x < x , it 

^ — rs — ru' 

follows that 



C^ < 2 EEE X X |Cov(H ,H ) 
~ r<s<u ^^ ^^ ^^ ^^ 



< 2 EEE x^ ICov(H ,H ) 
— ru ' rs ru 
r<s<u 



= 2EEju-r-l)x2jcov(H^3,H^^)|. 



Now (u-r-1) <_ N for all 1 £ r < u < N, and 



c 1 < 2NEE x" |Cov(H ,H ) 
2' - ^^^ ru' ' rs' ru' 



so 



28 



The two similar terms c_ and c. in (2.2.27) can also be 
bounded in this v/ay. Combining these bounds with the fact 
that all variance and covariance terms in (2.2.27) are 
uniformly o(l) we find 



Var[((T-(o^)-T(0))/a^^^] < B^, 



where B^ = (o(l)/N^a^ ^) (6N+1)ZZ x^ 

' r<s 



(3o(l)/N^a^) {6N+l)N^a^ 



= o (1) , as N -> ", 

and hence (2.2.26) is proved for u > . The proof is 
similar for u < . 

Proof of Theorem 2.2.2 (continued) ; 

We see by Lemma 2.2.3, Theorem 2.2.1, and Slutzky's 
theorem (Randies and Wolfe, 1979, p. 424) that under 
Hq: B^_2 = 0, 



T(0) + E((T(-a) )) 

^^— J N(2(3^)a)I(g) ,1) 

T,A 



as N -> «=. Using this result. Lemma 2.2.4 and another 
application of Slutzky's theorem shovz that under 

^0- ^1-2 " °' 



cffc — To«a »g- iT^ 



29 



T(-(^N^''''t,A t N(2(3^)wl(g) ,1) 



as N ^ ". Then by our remarks at the beginning of the proof 
of Theorem 2.2.2, we are done. 

Consider the Hodges -Lehmann estimate of 6,_^, say 6, ,, 
associated with the statistic T(b). If we let 
^1_2 = sup{B: T(3) ^0} and i^_^ = inf{B: T(3) < 0}, then 
we may define Q^_^ = (6^_2+eJ_2) /2 . We now give a theorem 
concerning the asymptotic distribution of 3t_,. 



Theorem 2.2.5 ; Under conditions I-IV, 



^^^x(^l-2"^l-2^ f N(0,(12l2(g)) 1) (2.2.28) 



as N ->■ ". 



Proof : 

■^®^ "n ~ '^Z (N'^^^) where w is a constant. Let P„{'} 



h 



denote the probability calculated when 3 = 3, and let < 
designate the standard normal cdf. From Theorem 2.2.2 it 
follows that 



lim P {T(0)£0} 



lim P {(T{0)/a ) - 2(3^)ajl(g) < -2 (3^) cal (g) } 



^a j WJT«i » IJ il im i U^j yi . W r-7 <ali 



30 



(-2(3 )ul(g)), or equivalently , 



lim P_ {T(C)£0} = * (2 (3^)0)1 (a)) . (2.2.29) 



By the definition of &^_^ and 3^_2 and using the fact that 



T(b) is nonincreasing in b, we have 



^1-2 ^ ^ "^ ^^^^ ^ ° "^ ^1-2 - ^' (2.2.30) 



^1-2 < ^ => T(b) < => bJ_2 ^ b. (2.2.31) 



Since by condition III the underlying distribution is 
continuous, it can be shown (using a proof similar to that 
of Theorem 6.1 of Hodges and Lehmann (1963)) that the 
distributions of e^_2 and ^2,-2 ^-^^ continuous. Thus 
(2.2.30) and (2.2.31) imply that 



Pp ii^ <h} = P {T(b)<0}, (2.2.32) 



P {8^ <b} =.P {T(b)<0}. (2.2.33) 

^1-2 -^ ^ Pi_2 



Since B^_2 = (1/2) (6^_2 + 65^_2) and 6^_2 < l^_2' it follows 
that 



^6,_2^^?-2<^> 1 ^B^_2^^1-2<^> 1 ^^_2^^1-2<^>' 



! ■»;» " . ii LJ« «M''*qw»Mwwww«»'wM«»«g°w5gT«»»°B wa J i ^ M U-. ■ im i g.'i^m. iii jB.ir^T—'rj 'i mf * m n iiiiiii.i . i i im o— — bm 



31 



and then substituting (2.2.32) and (2.2.33) into this result 
we have 

P {T(b)<0} < P {6 <b} < P. {T(b)<0}. (2.2.34) 



Because the distribution of T(B) when ^-,_2 = 3 is the same 
as the distribution of T(0) when 6,_p = (see proof of 
Theorem 2.2.1) it can be shown using the definition of B 
that the distribution of (^i_2~^i_2^ under &^_^ is the same 

A 

as the distribution of &-,-2 ^^®" ^i_2 "^ ^' Since the dis- 

A 

tribution of 3-J.-2 ^^ continuous and the limiting distribu- 
tion in (2.2.29) is continuous, we apply (2.2.34) to obtain 



lim P {N'a (6 -3 ) < to} 



= lim P-^{N^a^6T ^ <cj} 



lim P {3 < u^/ {N^'o ) } 



lim P {T(aj/(N'a^) ) < 0} (from (2.2.34)) 



= lim pQ{T(a3j^) < 0} 



-J«IJJl.J!'.]JieWW'^WSlJ-.'B i! im.BlMmMW.UM ^i^1l gW<g - «• "■ - ' ■ »i i..r nTOi.iia i. »« j i mi--. ca» 



32 



= lira P {T(0) _< 0} 

= §(2(3^) a)I(g)) (from (2.2.29)), (2.2.35) 



and the result (2.2.28) follows immediately. 

2.3 Large Sample Inference 
Using the results of the previous section, we now 
proceed to construct tests and confidence intervals for 
^l_2- These tests and confidence intervals follow from 
those presented by Sievers, now applied to the two line 
setting. 

Consider a test of H^ : B^_2 = &q against H^ : Q > 6 

based on the statistic T(eQ). Of course to test for 
parallelism of the regression lines we take g. = 0. Large 
values of T(3q) indicate the alternative H^ holds. For a 
given < a < 1, let z denote the 100 (1-a) percentile of 
the N(0,1) distribution. Then from Theorem 2.2.1, the test 
which rejects H if 



■"6o'''='t,a' \ 



is an approximate level a test. A two-sided test of 



H 







'2__2 = is derived by making the usual modifications. 



Since T (b) (2.1.5) is a nonincreasing step function of 
b with jumps at the slopes S = (Z -Z ) /x , 1 < r < s < N, 
a confidence interval for g can be constructed by 



33 



inverting the two-sided hypothesis test. This interval has 

endpoints that are properly chosen values from the set of 

slopes {S : 1 <_ r < s <_ N}. 

Let J(s) be the cumulative distribution function of a 

discrete random variable that takes the value S with 

rs 

probability x /x. . , where x.. = ZZ x . Then for each b 

r<s 

such that b7*S ,l_<r<s_<N, 



J(b) = (1/2) [l-(NT(b)/x..)] . (2.3.1) 

If the distribution of the Z ' s is continuous, then 
P{S^g = b for some l_<r<s_<N} = for each b and (2.3.1) holds 
with probability one. Therefore if we can determine 
constants t^ and t^ such that P{t, <T (6) <t2 } = 1 - a, then 



1 - a = P{^(l-(Nt2/x..)) < J(e) < J2(l-(Nt^/x..)) } 



= p{j;;;-^(^(i-(Nt2/x..))) ^ e < jj-(h{i-{i>it^/x. .)))}, 

(2.3.2) 



where J~ (u) = inf {s: J(s)>_u} and j"""" (u) = inf {s: J(s)>u} 
are inverses of J defined appropriately for our purpose. 
Thus 



[J^-^(%(l-(Nt2/x..))) , J^^(35(l-(Nt^/x..)) )) (2.3.3) 



--*-»»^»^«Wi-— .--T^CH';- -.^■.i-a''*>t:i>- »■■"--■ ~r<l.-JL. -..*■■■ T«JC— *».»f*'— ^-^ 



34 



is a 100 (1-a) percent confidence interval for 6, ^. 

If V7e use the asymptotic normality of T(3) in Theorem 
2.2.1 to determine t^ and t^ we find that 



^^~-^ ^'^-^a/2 <N^T,a/2^- • ) ) ' ^'J' ("s+z^/s (Na^^A/2x. . ) ) ) 

(2.3.4) 

is an approximate 100 (1-a) percent confidence interval for 
^1-2* ^^ ^^^ write this interval more explicitly in terms 
of the slopes S^^ as [S^'^^s'^) , where 



S = min{S^g: J (S^^) >%-2 , (Na /2x. . ) , l£r<s<N}, 

(2.3.5) 



and 



S = miniS^^: J {S^^)>h+z^^^{lio^ ^^/2x. .) , l<r<s<N}. 

(2.3.6) 



An alternative confidence interval follows directly 
from the asymptotic normality of &^_^ (Theorem 2.2.5). An 
approximate 100 (1-a) percent confidence interval for e._ is 
given by 

^1-2 ± 2,/2(2(3V)a^i(g))-l, 

where 1(g) is a consistent estimate of 1(g). Such estimates 
have been proposed by Lehmann (1963) and Sen (1966) . 



35 



2.4 Asymptotic Relative Efficiencies 
In this section we compare the Sievers-Scholz approach 
for testing H^ : e^_2 = b with the procedure due to Hollander 
and the procedure due to Rao and Gore. As noted in Chap- 
ter One, the Hollander and Rao-Gore procedures include exact 
confidence intervals for the slope difference 5-,_2' We will 
see in the next section that our proposed application of the 
Sievers-Scholz approach also includes exact confidence 
intervals for &2.-2' ^^^^^- these three exact, nonpararaetric 
procedures are the primary focus of the asymptotic relative 
efficiency comparisons presented here. Comparisons of the 
Sievers-Scholz test with the classical t-test based on least 
squares theory and Sen's (1969) test are also made. First 
we show how to construct each of the competing nonparametric . 
test statistics, along with a brief illustration of the 
rationale behind each one. We then describe a sequence of 
sample sizes tending to infinity for the purpose of 
computing the Pitman asymptotic relative efficiencies 
(pares) . Finally', we compute these efficiencies and compare 
their values assuming several different underlying error 
distributions. 

Since all three exact, nonparametric procedures can be 
expressed in terms of basic slope estimates for each line, 
we first define 

^irs = ^"is-^ir^/^rs' ^r " ^^s' (2.4.1) 



nMMi^«P9f« 



36 



the estimate of the slope of line i (i=l,2) resulting from 
the responses at x and x . Recall that in the previous 
sections we used a similar notation. 



3 = (Z -Z )/x , 
rs s r rs' 



to designate slope (difference) estimates computed from the 

differences Z., j=l,...,N. Of course S = S^ - S^ , and 
3 rs Irs 2rs 

the additional subscript indicating line 1 or 2 in the 
estimates in (2.4.1) should help to avoid any confusion. 
There will be {^^) = N(N-l)/2 of these estimates associated 
with each line. The slope estimates of line 1 are naturally 
independent of those of line 2, but the (2) slope estimates 
of a single line are not mutually independent. One way of 
motivating and com.paring the three exact, nonparam.etric 
procedures is to examine how they utilize these basic slope 
estimates in forming their test statistics. 



The Sievers-Scholz Statistic 

In the two line setting, the Sievers-Scholz statistic 

is appropriate only when the lines have common regression 

constants. As before, let x- < x„ < . . . < x„ denote the 

1 — 2 — — N 

regression constants. We can write the Sievers-Scholz 
statistic T(0) in terms of the slope estimates: 



T(0) =lEr x^3sgn(s,^.3-S2^^) 
r<s 



37 



Examining this representation we see that each line 1 slope 
estimate is compared v/ith the line 2 slope estim.ate 
resulting from the observations at the same regression 
constants. Thus all (p) slope estimates of each line are 
used, but an estimate from line 1 is compared only with the 
corresponding estimate from line 2. This results in („) 
comparisons across lines. Each comparison is weighted by 
the distance between the x's used in its construction. 



The Hollander Statistic 

Unlike the Sievers-Scholz statistic, the Hollander 
statistic is applicable even when the lines do not share 
common regression constants. However, use of the Hollander 
statistic requires a grouping scheme designating N/2 pairs 
of regression constants for each line. Assume N = 2k. When 
the x's are equally spaced, that is, when the regression 
constants for line i are 



X. T = L. + mc, m = 0, 1, ..., 2k-l, (2.4.2) 



i = 1, 2, for some constants L^ , L„, c^ , and c_, Hollander's 
grouping scheme pairs 



X. ^ with X. ^,, , m = 1, ..., k, (2.4.3) 

i,m i,m+k 



i = 1, 2. The first step in Hollander's procedure is to 
utilize the observations at each pair of x-values to 



38 



construct N/2 independent slope estimates of the fom 

(2-4.1) for each line. Hollander notes that under equal 

spacing (2,4.2) his grouping scheme (2.4.3) minimizes the 

variance of the slope estimates used among all grouping 

schemes that produce identically distributed slope 

estimates. Under unequal spacing of the regression 

constants, some ambiguity exists as to the choice of a 

grouping scheme. Hollander suggests devising a scheme that 

will yield pairs of x-values situated approximately a 

constant distance apart. 

Having computed N/2 = k independent estimates of the 

slope of line 1 and k independent estimates of the slope of 

line 2, the next step in Hollander's procedure is to 

randomly pair the slope estimates of line 1 with those of 

line 2. Let (S-j^^.^, ^2^^) designate one of the k pairs. For 

each pair the difference of the form S, - S^^ is 

Irs .2tu 

calculated. If we label these differences d^ , . . . ,± then 

1 k 

Hollander's statistic is computed as 



W = ^ ^m^^'^m^' (2.4.4) 

m=l 



where R^, is the rank of |d^,| among {|d^|: m=l,...,k}, and 
6 (a) = 1 if a > , otherwise; W is the Wilcoxon signed 
rank statistic computed using the k slope difference 
estimates as the observations. Writing a slope difference 



^•ft w p y M gw pn y 1 1 'im \\ t» m mi-ykm^ imm^Qr»\mo»mv ?9 e'^'* y*l»'SK~mr r 



39 



estimate d^ in terms of the underlying parameters, we see d 

has the form 






where x^^^ = x^^ - x^^, i = 1, 2. Since E^^ and E^^ are 
independent and identically distributed, it follows that the 
distribution of E^^' - E^^ is symjr.etric about zero. Clearly 
then , 



^is - ^ir ^2u - ^2t 
""irs ^2tu 



is symmetrically distributed about zero. Thus Hollander's 
approach does not require the same regression constants or 
equal spacing since the Wilcoxon distribution will apply 
under 6^_2 = regardless of the spacing or choice of 
regression constants. However, if the regression constants 
are the same and equally spaced, the asymptotic relative 
efficiency (ARE) to be presented in this section as Result 
2.4.2 indicates superiority of the Sievers-Scholz approach. 

Note that Hollander's approach does not use all 
available (2) basic slope estimates from each line. Instead 
a subset of N/2 independent line 1 slope estimates is 
selected from the {^) possible. A similar set is selected 
from the line 2 slope estimates. Each member of the first 



40 



set is compared with only one randomly selected member of 
the second set, resulting in only N/2 comparisons of slope 
estimates across lines. This is a sm.aller number of 
comparisons across lines than the (2) comparisons of the 
Sievers-Scholz method. Although there are dependencies 
among the Sievers-Scholz comparisons, the greater number of 
them would lead us to expect the Sievers-Scholz approach to 
be superior to Hollander's method when both are applicable. 
Again, the ARE Result 2,4.2 will confirm this. 

The Rao-Gore Statistic 

Let A(B) designate the set of N/2 slope estimates for 
line 1(2) used in constructing Hollander's statistic. Note 
that under equal spacing (2.4.2) with c^ = C2 = c and using 
Hollander's recommended grouping scheme (2.4.3), the 
distribution of the slope estimates in A differ from those 
in B only by a shift in location of 3-i_2 "= ^i ~ ^2' 



E, - E^ 
q - R + Is Ir 
^Irs " ^1 ^ kc 



c - B + ^2u - ^2t 
^2tu ^2 ^ kc 



Here x. , - x. = kc is the common distance between pairs 
i,m+k i,m 

of x's used in forming the slope estimates. Thus, under 
equal spacing, Rao and Gore proposed the Mann-Whitney- 
Wilcoxon statistic of the fcirm 



41 



" -{S.^JeA}{S2^^eB} ^ ^Slrs'^2tu)' (2.4.5) 



where y{a,h) = 1 if a > b, otherwise. It is clear that 
the Rao-Gore procedure would apply under any spacing and 
grouping scheme resulting in the same distance between all 
pairs of x's used to form slope estimates. Under such 
spacing, the Rao-Gore procedure eliminates the extraneous 
randomization needed by Hollander's procedure to pair the 
slope estimates of the two lines. 

We see the Rao-Gore procedure uses the same two sets of 

slope estimates as Hollander's approach. However all 

2 
possible (N/2) comparisons across lines are made. This 

leads us to expect that the Rao-Gore procedure will compete 
favorably with Hollander's method. 

Comparing the Rao-Gore technique with the Sievers- 
Scholz approach in the previous intuitive way is not as 
revealing. The Rao-Gore technique makes all possible 
comparisons across lines among two relatively small sets of 
independent slope estimates. The Sievers-Scholz approach 
makes only pairwise comparisons across lines, but uses all 
possible slope estimates. The sets of slope comparisons 
used in the two procedures are such that neither set is a 
subset of the other. We will see, however, that the ARE 
Result 2.4.3 indicates superiority of the Sievers-Scholz 
approach. 



■ in.ii i an i i_». i ., 



42 



The Sen (1969) Statistic 

Sen (1969) proposed a statistic for testing the 
parallelism of several regression lines. We introduce his 
statistic here in the two line setting. Note that Sen's 
statistic does not require common regression constants for 
the two lines as required by the Sievers-Scholz statistic. 
However we will assume common regression constants to ease 
the notation. Modifications needed to construct Sen's 
statistic in the more general case of different x's for the 
two lines will be obvious. Thus we assume the basic linear 
model (2.1.1) . 

Let (|) (u) be an absolutely continuous and nondecreasing 
function of u: < u < 1, and assume that ((> (u) is square 
integrable over (0,1). Let U^^^ < U^2) < ••• < U^^^ be the 
order statistics of a sample of size N from a uniform [0,1] 
distribution. Then define the following scores: 

Ej = E[MU(j))], (2.4.6) 



or 



Ej = <j)(j/N+l), (2.4.7) 

j = 1, , N. Define 



1 

** = / 0(u)du (2.4.8) 


and 



■■I.W .mimu g'ma^'Sf^ 



43 



2^2 2 

A = f ^ (u)du - (<i)*)^, (2.4.9) 





and consider the statistics 



V = [ E (x.-x)E ]/(An\ ), (2.4.10) 

3=1 -^ ij 



i = 1, 2, where R^^ is the rank of Y^. among Y., Y.^, ..., 
^iN' ^^^ observations of the ith line. Statistics such as 
(2.4.10) are used in the single line setting to test 
hypotheses about the slope. 

The function (j) (u) is called a score function. It is a 
function applied to the observed ranks of the Y's. The 
choice of cj) can be made to achieve desirable power 
properties of a test based on (2.4.10). These properties 
depend in part on the underlying error distribution. A 
general discussion of score functions is beyond the scope of 
this work. We will discuss the score function here only to 
the extent needed to clearly present Sen's statistic. 

Assume F, the underlying error cdf of the assumed model 
(2.1.1) , is absolutely continuous and 

I*(F) = / [|^]2dF(x) < ». (2.4.11) 



wp <i"'ji^wj: iT ^p-. g^ BRm a 



44 



Then we reserve the symbol ^ (u) for the following score 
function: 



,(u) = _ fMZ^iluU , < u < 1. (2.4.12) 

f(F ^(u)) 



It can be shown. 



1 
/ 1'(u)du = 0, and 




1 2 
/ 'P (u)du = I*(F) . 




The score function Y (u) has been shown to have certain 
desirable properties when applied to the two-sample location 
problem (Randies and Wolfe, 1979, p. 299). 
We also define 



1 

Pi'^r<i>) = [/ 'i'(u)^(u)du]/[A^I* (F)]^, (2 4 13) 





which can be regarded as a measure of the correlation 
between the chosen score function <p and the optimal one C?) 
for the error distribution being considered- The expres- 
sions p(V,(j)) (2.4.10) and I* (F) (2.4.11) will appear in the 
development of the statistic based on Sen's (1969) work. 



45 



We now define Sen's statistic. Let 



Vidi+bx) (2.4.14) 



denote the value of V^ (2.4.10) based on Y. + bx , Y. + 
bx2, ..., Y^^ + bx^. Define 



V = (V^ + v^)/2. (2.4.15) 

Assuming H^ : &^_^ = o, let 6* denote the Hodges-Lehmann 
estimate of the common slope of the two lines based on V. 
Define 

Vi = V.(Y.-3*x), (2,4.16) 

i = 1, 2. Then Sen proposed the statistic 



. _ 2 .2 

^ ~ -l.^i (2.4.17) 



to test Hq: B^__2 = against H^ : &^_^ ^ 0. 

The statistic L is a quadratic form in the V., i = l, 2. We 

now give an intuitive motivation for the form of L. 

The statistic V^ is the value of V. based on the 
observations 



^il - ^^-^^1' ^12 - ^*-2 ^iN - e*-N- 



46 



Under H^ : &^_^ = , the 2 lines are parallel and g* is an 
appropriate estimator of the corrjr.on slope. In that case, 
the transformed observations behave essentially as random 
errors fluctuating about zero. Then L, being the sum of 
squared random errors with mean zero, has an asymptotic 
central chi-squared distribution, and a test of H may be 
based on L using this asymptotic null distribution. Under 
H^: 6^_2 7^ 0, the estimate B* is not appropriate since the 
two slopes are not the same. The transformed observations 
will not, in general, have mean zero. Hence the value of L 
will be larger than expected under the null, and the use of 
the null, central chi-squared critical values will tend to 
lead to rejection of H . 

PARE Specifics: Alternatives, Regression Co nstants , 
Sequence of Sample Sizes ' 

In computing the PAREs, we assume a sequence of 
alternatives to the null hypothesis H : g = o specified 
^y ^N" ^1-2 " ^1-2^^-^ " w/N'a^ as in Theorem 2.2.2. 
However, following Hollander (1970) , we consider only 
equally spaced common regression constants resulting from 
setting c^ = c^ = c, L^ = L^ = L in (2.4.2), with n 
observations per line at each of the 2k x-values. Clearly 
then, N = 2kn, a^ = c^ (4k^-l) /12 , and 

H^: B^_2(N) = 6^_2(2kn) = [o)/ (2kn) ^] [2 (3^) /c (4k^-l) ^] . 

(2.4.18) 



47 



The PARES of the test procedures will be derived under two 
different schemes for allowing the sample size N = 2kn to 
tend to infinity: 



Ca.se 1; Let C be a positive finite constant and let c = c 
such that kc, ^ C as k ^ ». Consider n fixed, ^ 

Case_2: Let c be constant. Consider k fixed, n -> «> , 
resulting in a PARE in terms of k. Then let 
k ■> <» in this expression. 



Case 1 essentially allows the number of distinct 
regression constants (2k) to tend to infinity over a fixed 
region of experimentation. Case 2 considers the number of 
distinct regression constants fixed while the number of 
replicates (n) tends to infinity. The efficiency 
expressions derived under case 1 and case 2 will be seen to 
be identical. In view of (2.4.18), the rate at which the 
sequence of alternatives converges to the null is inversely 
proportional to the square root of the sample size, N = 2kn, 
for both case 1 and case 2 . 

Before proceeding to derive the various PARE's, we 
state a theorem due to Noether (1955) as presented by 
Randies and Wolfe (1979, p. 147). 



Theorem 2.4.1 (Noether ' s Theor em) . Let {S ,.,} and 

~ — • n (x) 

{T^,(i)} be two sequences of tests, with associated 
sequences of numbers { Pg (n (i) ) ^^^ > ' ^ '^T (n' (i) ) ^ ' ^ ^ ' 



48 



2 2 

^^S(n(i)) ^^^ ^' ^^^ ^"T(n(i)) ^®^ ^' ^^^ satisfying the 

following Assumptions A1-A6: 



M. 'n(i) - ^S(n(i))(Qi) .^^ ^n-(i) " ^T(nWi))<^) 
''S(n(i))(^^ ^T{n'(i))^'i^ 



have the same continuous limiting (i ^ ») distribution with 
cdf H(.) and interval support when 6. is the true value of 

•it: 



A2. Same assumption as in Al but with 0. replaced by 6 
throughout. 



1^00 (J ... (ft \ -i -i.m ^ /n \,— 1 



^S(n(i))(Qo) i-^- ^T{n'(i))<%^ 



A4. ^ 



de f'^S(n)(^^] = ^s(n) <^) ^"^ 



fe f^T(n')^^^^ = ^T(n') ^^^ 



are assumed to exist and be continuous in some closed 
interval about 9 = e^, with V^^^^ie^) and y'(^.)(eQ) both 
nonzero. 



A5. 



li^ - S(n(i))'-i^ ^ .^ ^ T(n'(i))'"i^ 



S(n{i)) ^^0' "^"' ^ T{n'(i)) ^'O' 

^6 lim ^'s(n) ^Qq) 

^'^' n-°° 2 T = K and 

K(„)'^o"' 



49 



lim ^'T(n') ^^o^ 



■ , , 2 J- ~ ^T ' 

where K^ and K^ are positive constants, called the 
efficacies of the tests based on S^ and T^, respectively. 
Then the PARE of S relative to T is 



PARE (S,T) .= -\ . (2.4.19) 

^T 



Proof ; 

See Randies and Wolfe (1979, p. 147). 

Note that assuming the equally spaced regression 

constants described just prior to (2.4.18) it follows that 

conditions I and II hold under either case 1 or case 2. 

Hence we need only explicitly assume conditions III and IV 

to apply Theorem 2.2.2 which establishes the asympototic 

distribution of the Sievers-Scholz statistic under H . We 

N 
now present the PAREs, 

PARE (Sievers-Scholz, Hollander ) 

Result 2.4.2; Assume the sequence of alternatives {H } 
(2.4.18) and the equally spaced regression constants 
described just prior to (2.4.18). Also assume conditions 
III, IV, and I(h)<". Then the PARE of the Sievers-Scholz 
statistic T(0) with respect to Hollander's W under case 1 or 
case 2 is 



50 



2 
PARE(Sievers-Schol2, Hollander) = ^[ '^ ^ ^^^ ] . (2.4.20) 

■^ 21^ (h) 



Verification of Result 2.4.2 ; We apply Noether's 

theorem. Assumptions A1-A6 must hold for T(0) and W. For 

2 

T(0)/a , Theorem 2.2.2 establishes Al and A2 with the 

standard normal limiting distribution and standardizing 
constants (suppressing the various subscripts and taking 

e=6^_2) 



y (9) = 4knl (g) 6 , and 
a^{8) = 8kn/[c^ (4k^-l)] . 



Then A3 , A4 , and A5 follow immediately from the form of 
these standardizing constants. Assumption A6 holds and 

efficacy (T(0)) = lim y ' (0) / [ (2kn) '^o (0) ] 

Itlig) case 1, 

(2.4.21) 
c(4k^-l)^I(g) case 2. 

We note that W is the Wilcoxon signed rank statistic 
computed using nk independent random variables with the 
distribution of 



^l"^2"^3^^4 
'1-2 ^ k^ 



51 



As in Randies and Wolfe. (1979, p. 165-166), the assxunptions 

A1-A6 can be validated using the equivalent statistic 

nk 
W/ (^ ■") , in which case 



y(6) = (2/(nk-l)) [1-H (kc (-6) ) ] 

+ 1 - /H(kc(-t-2e))dH(kct) , 
o^(e) = l/(3kn) , 

the limiting distribution in Al and A2 is the standard 
normal , and 



efficacy (W) = lim y ' (0) / [ {2kn) ^a (0) ] 



N->-<» 



6 ^CI (h) case 1, 



J, 
6 ^kcl (h) case 2 . 



(2.4.22) 



Then (2.4.20) follows immediately from (2.4.19) of Noether's 
theorem. 

The first four rows of Table 1 show the value of the 
PARE (2.4.20) when the error distribution F is uniform, 
normal, double exponential, and Cauchy. The distributions 
are listed in order of the increasing heaviness of their 
tails, using the measure 



F-^.95) - F-^.50) ^ ^.^^_23) 



F ^ (.75) - F~^ (.50) 



52 



Table 1. PARE (Sievers-Scholz , Hollander) and Heaviness of 
Tails for Selected Error Distributions. 



Distribution 



Uniform 

Normal 

Double Exponential 

Cauchy 



PARE 



1.29 
1.33 
1.48 
2.67 



Heaviness of Tails 



1.80 
2.44 
3.32 
6.31 



CN(e,a^,5,.9) 

CN(e,a^,5,.8) 

CN(e,a^,5,.7) 

CN(e,a^,5,.6) 

CN(e,a^,5,.5) 

CN{e,a^,10,.9) 

CN(e,a^,50,.9) 

CN{e,a^,100,.9) 

CN(e,a^,500,.9) 

CN(e,c^,1000,.9) 



1.71 
1.94 
1.98 
1.86 
1.70 
2.13 
2.83 
2.96 
3.07 
3.08 



2.82 
4.10 
5.16 
5.34 
5.02 
3.07 
3.68 
3.92 
4.51 
4.78 



1) Heaviness of Tails = 



F"^(.95) - f"^(.50) 
f"^(.75) - f"^(.50) 



2) CN(e,a ,ic,t) represents the mixture of two independent 

normals. The first, chosen with probability x, has mean 

2 
6 and variance a'. The second, chosen with probability 



1-T, has mean 



2 2 
and variance k a 



53 



defined by Crow and Siddiqui (1967), where F~-'-[F(t)] = t. 
It is clear that for these four distributions, the PARE of 
the Sievers-Scholz statistic with respect to Hollander's 
statistic increases with increasing heaviness of tails. To 
see whether this behavior persists for other distributions, 
we examined the PARE (2.4.20) for a variety of contaminated 
normal distributions defined as follows. 

Let Z = X with probability t and Z = Y with probability 
1-T, where X is N(e,a'") and Y is independently N(e,K^a^). 
Then we say Z is distributed as a scale contaminated com- 
pound normal which we designate by CN (6 , a^ , k , t) . VJhen F is 

2 
the CN(e,a ,k,t) distribution, straightforward computations 

yield the following formulas for 1(g) and 1(h): 



,4, 4-i 



,4> 4-1, T ,1 

(.)t (1-t) 



Kg) = -T^— J {-^^- Y— r^ ' (2.4.24) 

TT^a ,tr, [2(i(K^-l)+4)] ' 



, g (J)tS-Ni-t)^ 

1(h) = -^^^— ; {-^ J-} . (2.4.25) 

^'a /I [2(i(K^-l)+8)] ' 



In addition to the four common error distributions. 
Table 1 shows the value of the PARE (2.4.20) for several 
contaminated normal error distributions along with the 
heaviness of the tails of these distributions. The formiulas 
(2.4.24) and (2.4.25) were used to compute these PAREs while 
iterative techniques provided the heaviness of tails of the 



54 



various contaminated normal error distributions. Figure 1 
shows a plot of the PARE (2.4.20) against the heaviness of 
tails for all the error distributions in Table 1. The main 
purpose of Table 1 and Figure 1 is to show the PARE (2.4.20) 
of the Sievers-Scholz statistic T(0) to Hollander's W is 
greater than one over a wide range of underlying distri- 
butions. Secondarily, we notice in Figure 1 that although 
no exact relationship exists, distributions with heavier 
tails tend to show higher PAREs. 

Under case 2, using the efficacies in (2.4.21) and 
(2.4.22), it follows that the PARE of the Sievers-Scholz 
statistic to Hollander's statistic before allowing k to tend 
to infinity is 



4k^-l r I^(q) i 

2 '■ 2 ^ ' 
3k 21 (h) 



Thus we see that the PARE is increasing in k, with a minimum 

2 2 . . 

value of I (g) /2I (h) for k=l . This is likely due to the 

fact that Hollander's procedure uses a decreasing proportion 

of the available slope estimates as k ->• «>. We have chosen 

to present our results after allowing k ->- °° for ease in 

interpretation and to avoid the need for a separate 

discussion of case 1 and case 2. 

PARE (Sievers-Scholz , Rao-Gore) 

Result 2.4.3 : Assume the sequence of alternatives {H } 
(2.4.18) and the equally spaced regression constants 



Figure 1. Plot of PARE (Sievers-Scholz , Hollander) versus 
heaviness of tails for selected error 
distributions. 



1) Heaviness of Tails = 



F -^(.95) - f"^(.50) 
F~^(.75) - f"^(.50) 



2) U = Uniform 
N = Normal 

E = Double Exponential 
C = Cauchy 

2 

3) CN(e,a ,k,t) represents the mixture of 2 independent 

normals. The first, chosen with probability t, has mean 
e and variance a . The second, chosen with probability 
1-T, has mean 6 and variance k^o^. 



T^ = CN(9,a^,5,.9) 
T^ = CN(0,a^,5,.8) 
T3 = CN(e,a^,5,.7) 
T^ = CN(e,a^,5,.6) 
T^ = CN(e,a^,5,.5) 



K^ = CN(e,a ,10,.9) 
^2 = CN(e,a^,50,.9) 
K3 = CN(e,a^,100,.9) 
K^ = CN(e,a^,500,.9) 
<5 = CN(6,a^,1000,.9) 



PARE I 
J. 5 +. 



56 



2.5 ■»• 




Kit — K5 



T4 



1.5 + 



0.5 + 



TS 




Heaviness of TpIIs 



-'^F'»wi»w^w^gWB»»»W'^Ba8waic«iMBaw »MML i ai « ». i T ii|ii tu-uj- i M i[i^ j» l>i iaij i j ^'MW[;^B^ 



57 



described just prior to (2.4.18). Also assume conditions 

III and IV. Then the PARE of the Sievers-Scholz statistic 
T(0) with respect to the Rao-Gore statistic U under case 1 
or case 2 is 

PARE (Sievers-Scholz, Rao-Gore) = 4/3. (2.4.26) 

Verification of Result 2.4.3 ; As in Randies and Wolfe 
(1979, p. 170-171), the assumptions A1-A6 of Noether's 
theorem can be validated for the statistic U + (nk (nk+1) /2) , 
equivalent to U, in which case 



y(e) = (nk)^/G(kc(t+e))dG(kct) + (nk (nk+1) /2) , 
a^(e) = n^k^/6, 



the limiting distribution in Al and A2 is the standard 
normal , and 



efficacy (U) = lim n ' (0) / [ (2kn) ^a (0) ] 

3 '01(g) case 1, 



(2.4.27) 



3 ''kcl (g) case 2. 



We showed A1-A6 hold for T(0) in the previous verification 
and gave the efficacies of T(0) in (2.4.21). Thus (2.4.26) 
follows imiD.ediately from (2.4.19) of Noether's theorem. 



® 



58 



PARE(Sievers-Scholz, Classical Least Squares) 

Result 2.4.4 : Consider the classical least squares 
theory t-test, as specified by Hollander (1970) . Assume the 
sequence of alternatives {E^} (2.4.18) and the equally 
spaced regression constants described just prior to 

(2.4.18). Also assume conditions III, IV, and o 

2 
a = Var(E^)<«>. Then the PARE of the Sievers-Scholz 

statistic T(0) with respect to the classical t-test 

statistic under case 1 or case 2 is 



PARE (Sievers-Scholz, classical) = 24a^[I^(g)]. (2.4.28) 

Verificat ion of Result 2.4.4 ; Let B^ and ^2 denote the 

least squares estimates of 6^ and B^' respectively, and let 

2 
s be the residual mean square error (see (3.1) of Hollander 

(1970, p. 389)). Then the form of the t-test statistic used 

to test Hq: Bj^_2 = under the specified equally spaced 

regression constants is 



e - 6 

^ - 1 2 

t - 2 • (2.4.29) 

r 12s ,h 

2 2-' 
nkc (4k^-l) 



Using the fact that s is a consistent estimate of a^ under 
B^^ and a discussion similar to that in Randies and Wolfe 
(1979, p. 164-165), it follows that assumptions A1-A6 of 
Noether's theorem hold for the statistic t with 



iJiii II WH»*Wiui I " wii ■ ■ "I, j-ionwif ij .11 iiiH» 



59 



y{e) = ^- ' ^"'^ (2.4.29) 

r 12a^ .h 

^ 2 ? ^ 

nkc (4k -1) 

a^(e) - 1. 



The limiting distribution in Al and A2 is the standard 
normal and 



efficacy (t) = lim y ' (0) / [ (2kn) 'a (0) ] 



[ — 5-] case 1, 

6a 



, c^(4k^-l) ,^ 

[ ^ 5—^] case 2. 

24a^ 



(2.4.30) 



A1-A6 hold for T(0) as shown in the verification of Result 
2.4.2 with the efficacies given by (2.4.21). Thus (2.4.28) 
follows immediately from (2.4.19) of Noether's theorem. 

The efficiency (2.4.28) is identical to the familiar 
PARE of the two-sample Wilcoxon-Mann-Whitney test with 
respect to the two-sam.ple normal theory t-test when the cdf 
of the underlying error distribution is G. Hence, the 
PARE (Sievers-Scholz, classical) > 0.864 for all G (see 
Hollander (1970) for a proof that in this case the inequal- 
ity is strict). Also, this PARE equals 0.955 when G is 
normal and is greater than one for many non-normal G. 



60 



PARE (Sievers-Scholz.Sen) 

Result 2.4.5: Assume the sequence of alternatives {H,^} 
(2.4.18) and the equally spaced regression constants 
described just prior to (2.4.18). Also assum.e the following: 

1) Conditions III and IV. 

2) I* (F) < ~ as in (2.4.11), where F is the under- 
lying error distribution. 

3) The score function ({) (u) is an absolutely continuous 
and nondecreasin.g function of u, < u < 1, that is 
square integrable over (0,1). 

Let pCy,!})) be defined as in (2.4.13). Then the PARE of the 
Sievers-Scholz statistic T(0) with respect to Sen's (1969) 
statistic L under case 1 or 2 is 



PARE (Sievers-Scholz, Sen) = J"^ -*• ^^^ ' (2.4.31) 

p^ (¥,<!)) I* (F) 



Verification of Result 2.4.5 : 

Sen's statistic L has an asymptotic (N->-<») chi-squared 

distribution under H (Sen, 1969, p. 1676). Using results 

_* 
from Sen's work we define a statistic, V , that has an 

asymptotic normal distribution under H and whose square 

(multiplied by a constant) is asymptotically equivalent to L 

_* 

under H . Since the square of V (multiplied by a constant) 

has the same asymptotic distribution under H as L, we use 

_* 

V in applying Noether's theorem to derive efficacy 



?^s^sa>B9!gi^~ap;> i i -i ji gi pt* ' »n, <, !■ hjiuhii 



61 



expressions for Sen's test and the PARE (Sievers-Scholz , 
Sen) . 



Let B^ denote the Hodges-Lehmann estimate of 3, based 
on V^ (2.4,10), i=l,2, and let 6* = 1/2(6* + 6*)* From 
equations (3.10) and (3.20) in Sen (1969, p. 1673, 1675) it 
follows that 



a = N'a (6 -3*) = 0^(1) and 
^ J. p 



as N^" under H^, i=l,2. Using these definitions of a and b 
and the notation (2.4.14) we apply Leimna 3 . 2 in Sen (1969, 
p. 1674) , which is given in Chapter Three of this work as 
Lemma 3.4.3, yielding 



V.(Y.-B X) - V.(Y.-6.x) 



= N'a^(6*-3*)p(¥,(|,) [I*(F)]^ + o (1) 



as N-^- under H^. Equation (3.22) in Sen (1969, p. 1675) 
states 



|Vi(Y.-e*x)| = o (1) 



•^'"'''Wl f .l ii . ' l"_ l «.>--'_P_» i|«l . l lv « LM i«» 



62 



and so, applying this to the previous result. 



^i " Vi(Ii-^*2S) 



= N'a^(S*-6*)p(Y,<(,) [l*(F)]'^ + o (1) 



as N->co under H^. Applying equation (3.19) in Sen (1969, 
p. 1675) , 



N^a^{6*-3*) = o (1) , 



we have 



^i " N'a^(B%B*)p (T,^) [i{F)]^ + o (1) 



as N^-<» under H^. It follows that 



63 



2 

L = p^(Y,<f)I*(F)NaJ E (e*-3*)^ + o^(l) 

^ i=l ^ P 



= p^(^,(j))I*(F)Na^%(3*-32)^ "^ °p^^^ 



as N->» under H„. We define the statistic 
N 



-* ■ H. * * 



V = (N/2) cr^(3-^-62) (2.4.32) 



and note that the asymptotic distribution of 

2 * -* 2 
p ('!',(()) I (F) (V ) under H is the same as that of L. From 

Lemma 3.4 in Sen (1969, p. 1676), 



p(^,<i>) [I*(F)]^[V*-(N/2)'^a^(6^-S2)] t N(0,1) 



as N-»-« under H . Thus assumptions A1-A6 of Noether's 

_* 
theorem hold for the statistic V with 



y(0) = [c(4k^-l) V(2(3'))] G, and 



o^ie) = [p^(^,<^)i^ (F)] ^ 



64 



The limiting distribution in Al and A2 is the standard 
normal and 



efficacy (V ) = lim y ' (0) / [ (2kn) ^o (0) ] 



h, ,.% 



[Cp(T,.^) (I (F))^]/6^ case 1, 

(2.4.33) 

.[c {4k^-l) ^p (¥,<(,) (I* (F) ) '^j / [2 (6^) ] 



The assumptions A1-A6 hold for T{0) as shown in the 
verification of Result 2.4.2 with the efficacies given by 
(2.4.21). Hence (2.4.31) follows immediately from (2.4.19) 
of Noether's theorem. 

To evaluate the expression (2.4.31), we first note the 
following, which results from the definition of A^ in 
(2.4.9) and q(1 ,^) in (2.4.13) : 



P^('i',<).)I*(F) = [ / ¥(u)(j,(u)du]2/A^ 



^ 2 1 

= [ / >l'(u)(|)(u)du]"^/[ / [ Mu) -<()*] ^du] . 



(2.4.34) 



Suppose we assume Mu) = u, < u < 1. Scores resulting 
from this choice of score function are called Wilcoxon 



65 



scores. Direct computations show that in this case 



1 2 

/ [(j)(u) - ^*]^du = 1/12. (2.4.35) 



Using a derivation similar to one in Randies and Wolfe 
(1979, p. 308) we show that when ^ (u) = u, the numerator of 
the right-hand side of (2.4.34) equals I^(f): 



/ Y(u)(|) (u)du = - / u[f' (F "-(u) )/f (F ^(u))]du. 


(2.4.36) 

Let t = F (u) , resulting in 



1 Ul) 

/ 4'(u)(i. (u)du = - / F(t)f'(t)dt, (2.4.37) 
UO) 



where F(?(p)) = p. Now let u = F(t), dv = f'(t)dt and apply 
integration by parts to (2.4.37): 



^ Ml) ^^^) 2 

/ ^(u)((.(u)du = - {[F(t)f (t)]^ ; - / f^(t)dt} 

^^^> UO) 

= [F(UO))f (5(0))] - [F(C(l))f (UD)] + / f^(t)dt. 

5(0) 

(2.4.38) 



If we assiime the support of F is [a,b] where a < b and 

f(x) ^-Oasxfborx + a, then we take 5(0) = a, 5(1) = b 

in (2.4,38) and 



66 



1 b 

/ Y (u) (}) (u) du = / f^ (t)dt = 1(f) . (2.4.39) 

a 



As Randies and Wolfe (1979, p. 313) state, this same form 
(2.4.39) can be obtained under more general assumptions. 

Hence substituting (2.4.35) and (2.4.39) into (2.4.34) 
it follows from (2.4.31) that under ^ (u) = u, H , and 
assumed regularity conditions, 



2 
PARE(Sievers-Scholz, Sen) = ^^ ^^^ . (2.4.40) 

I (f) 



Table 2 gives values of the PARE (2.4.40) for four 
common error distributions. We see that when compared to 
the Sen statistic using Wilcoxon scores, the Sievers-Scholz 
statistic achieves a PARE close to (or equalling) one for 
error distributions having light to moderately heavy tails 
(uniform, normal, double exponential) . However the 
Sievers-Scholz statistic has poor PARE under the Cauchy 
distribution which has very heavy tails. That the 
asymptotic performance of Sen's test is better than other 
tests is not surprising since Sen's test is the rank test 
that maximizes the efficiency relative to the likelihood 
ratio test. However, Sen's test not only requires iterative 
calculations, but is distribution-free only asymptotically. 
More discussion of the relative merits of Sen's test and the 
Sievers-Scholz testd will be given in Section 2.6. 



67 



Table 2. PARE (Sievers-Scholz , Sen) for Selected Error 
Distributions Assuming (j) (u) = u. 



Distribution 



Uniform 

Normal 

Double Exponential 

Cauchy 



PARE (Sievers-Scholz, Gen) 

0.89 
1.00 
0.78 
0.50 



Summary of PARE Results 

In conclusion, PAREs derived under equally spaced 
regression constants favor the Sievers-Scholz approach over 
the other two exact, nonparametric competitors due to 
Hollander and Rao and Gore. Specifically, the PARE of the 
Sievers-Scholz statistic with respect to Hollander's 
statistic is greater than one over a wide range of 
underlying error distributions. Indeed this efficiency 
frequently exceeds two. Even more interesting is the fact 
that the PARE of the Sievers-Scholz statistic with respect 
to the Rao-Gore statistic is 4/3 for all error distributions 
(subject to certain regularity conditions required to derive 
the PARE) . The Sievers-Scholz statistic achieves the 
familiar PARE (2.4.28) when compared with the classical 
least squares theory t-test. Although the PARE of the 
Sievers-Scholz statistic with respect to Sen's statistic is 



68 



less favorable under the heavily tailed Cauchy distribution, 

we noted that Sen's test is distribution-free only 
asymptotically. V7e discuss these two methods further in 
Section 2.6. 

Note that under equal spacing, the PARE of the Sievers- 
Scholz statistic to the Theil-Sen statistic (2.1.4) dis- 
cussed in Section 2.1 is one (Sievers, 1978). Thus all of 
the previous ARE comparisons also apply when using the 
(zero-one weighted) Theil-Sen statistic in place of the 
Sievers-Scholz statistic. The advantage of the Sievers- 
Scholz approach over that of Theil and Sen will appear in 
the Monte Carlo comparisons under unequal spacing of the 
regression constants discussed in Section 2.6. 

2.5 Small Sample Inference 
Since the test of parallelism and the confidence 
intervals for g^^^ presented in Section 2.3 depend on 
asymptotic theory, they are generally only applicable with 
moderately large samples. In this section we discuss exact, 
distribution-free tests of H^ : g^^^ = and corresponding 
exact confidence intervals for the slope difference, B 
These tests continue our basic approach of applying the 
method of Sievers and Scholz to the two line setting, again 
assuming comm.on regression constants. 

Specifically, we discuss two related techniques. The 
first utilizes the exact distribution of the Sievers-Scholz 
statistic TiO) with the optimum weights, a = x , under 






69 



the null hypothesis H^ : 3-,_2 = 0. This null distribution 
depends on the chosen regression constants, and hence must 
be recalculated for each design. The second technique is a 
straightforward application of the Theil-Sen approach to the 
two line setting, as discussed in Section 2.1. Since the 
required null distribution is essentially that of Kendall's 

o 

tau, tabled critical values are readily available for small 
sample sizes. Thus the second technique has favor under 
sample sizes sm.all enough to discourage the use of 
asymptotic results, yet too large to allow the computation 
of the null distribution of T(0) required by the first 
technique. 

Let us now discuss in detail the two small-sample 
techniques we have proposed. First, recall the basic linear 
model (2.1.1) we have assumed, the resulting distribution of 
the differences Z. given by (2.1.3), and the representation 
of T(0) in terms of the ranks of the Z's given by setting 
b = in (2.2.1) : 



N 
T(0) = (2/N) Z [Rank(Z.)x ] - (N+l)x. (2.5.1) 

j = l ^ ^ 



Suppose that for line i (i=l,2) the underlying errors, E. ., 
j=l,...,N, are independent and identically distributed 
(i.i.d.). The distribution of the line 1 and line 2 errors 

need not be the same, but assume these two distributions are 
continuous. Let ^ denote the set of N! permutations of 



70 



(1,2,...,N). Then under H^ : 3-|^_2 = 0, the Z's are i.i.d. 
and hence the vector of ranks of the Z's is uniformly 
distributed over ^^ (Randies and Vvolfe, 1979, p. 37). 
Consequently, in view of the representation (2.5.1), the 
null (B^_2 = 0) distribution of T(0) is uniformly 
distributed over the N! values of T(0) obtained by permuting 
the ranks of the Z's among themselves. For example, if for 
r = (r^,. . . ,r^) e|^, 



N 
t„ = [ (2/N) E (r.x.)] - (N+l)x , 
- j=l ^ ^ 



then the distribution of T(0) under H. : g = is given by 

Pq {T(0)=t^} = 1/N!, rti^, (2.5.2) 

where, as before, ^^i ' } denotes the probability calculated 
assiiming B^^^ = te Having tabulated this null distribution, 
a test of H- is conducted as follows. 

If t^ is a constant determined from (2.5.2) such that 
Pq {T(0)>_t^} = a, then the test of H^ : &^_^ = against 
Hq: B^_2 > which rejects H„ if 



T(0) 1 t^ (2.5.3) 



is an exact level a test. Rather than simply state whether 
a test has accepted or rejected a null hypothesis at a 



i MU i u 'ww LHJ ^ - ' j ^ tig ii j iw ^i^ tt Mw u ' L iii ^T, v ftai ri ^Sf* 



71 



particular level, one might wish to report the attained 
significance level, the lowest significance level at which 
the null hypothesis can be rejected with the observed data, 
If we observe T(0) = t(0), then the exact test (2.5.3) has 
an attained significance level of 

p = [ (number of r in P) /N! ] , 

where P = {re^,: t(0)£t^}. Two-tailed tests can be 
obtained by the usual modifications. 

Lst t^ -2 be a constant defined analogously to t : 



^0 ^-^a/2^^(0)^^a/2> = 1 " - 



Applying the argximent and notations used to derive (2.3.3), 
it is easily seen that 



[j;^(%(l-(Nt^/2/^--)))' jZ^(^a+i^t^^^/x..)))) 

(2.5.4) 



is an exact 100 (l-a) percent confidence interval for 6. 
In terms of the slopes S^^ = (Z -Z ) /x , 1 j< r < s <_ N, 
we can write this interval as [S""^, S^) , where 



Sg = min{S^g: J (S^^) >% (1- (Nt^^2/^- • ) ) ' l<r<s<N}, 



and 



MB'.m^.r^Tl'^igi l 



72 



Sq = min{S^^: J (S^^) >h {1+ (^t^^^/x. .) ) , l.<r<s<N} . 

(2.5.5) 

Computation of a null distribution based on permuta- 
tions of the observations (or ranks) such as the one in 
(2.5.2) has become possible under small sample sizes with 
the speed of today's computers. At the University of 
Florida we found that the necessary calculations were 
feasible when there were N=8 or fewer regression constants 
per line. In this case the entire null distribution of T(0) 
could be tabulated at a cost of about $2.00 to the user. We 
were accessing a system operating an IBM 3081 with MVS/XA 
and an IBM 3033 with OS MVS/SP JES2 Release 3. With more 
than 8 regression constants per line, the cost of tabulating 
the null distribution becomes prohibitive (at least $30.00 
when N=10, for example). However, in dealing v/ith similar 
problems, Pagano and Tritchler (1983) and Mehta and Patel 
(1983) give algorithms that greatly reduce the amount of 
computation involved. Although their results do not apply 
directly to this problem, we anticipate the exact test 
(2.5.3) and computation of the confidence interval (2.5.4) 
will soon become feasible for larger sample sizes due to the 
development of similar efficient algorithms and the steadily 
increasing speed of computer hardware. 

At the present time we could resort to estimation of 
the null distribution (2.5.2) of T(0) based on a random 
sample of permutations of the observed rank vector. Boyett 



■ "" i^u f^ gJC .1 1 ii a^ i qm BBj rL P'!WqiMWi^-*?j g, 'g"' . » ajt'j » i ■■ wn nn mngi w em w * t atm m nuiim 



73 



and Shuster (1977) and Ireson (1983) report good approxima- 
tions resulting from the use of such a sampling of permu- 
tations to approximate a permutation distribution. However, 
there is a possibility that approximate procedures such as 
these suffer a loss of power due to the restricted sample 
space (Dwass, 1957) . Also, there is the problem of specifi- 
cation of the number of permutations that must be sampled to 
achieve adequate approximations of the null distribution. 
Clearly, more study is needed before this technique can be 
recommended without reservation. 

Another method of overcoming the computational problem 
associated with the exact test under larger sample sizes is 
to replace the Sievers-Scholz statistic by the Theil-Sen 
statistic based on the differences Z . , as described in 
Section 2.1, Recall that this consists of using the weights 
^rs " sgn(x^^) in the expression for T (b) given by (2.1.2). 
The null distribution is essentially that of Kendall's tau, 
which has been tabulated for many values of N. These tabled 
critical values and a precise specification of the Theil-Sen 
test and confidence interval are given in Hollander and 
Wolfe (1973) . This second approach is appropriate when an 
exact, distribution-free technique is desired but the number 
of regression constants per line is too large (N>8, at our 
facility) to allow complete enumeration of the null distri- 
bution (2.5.1) of T(0). If the regression constants are 
highly unequally spaced, the method based on Sievers-Scholz 
procedure probably has greater power, but the results of a 



74 



simulation study discussed in the next section indicate that 
in many situations the difference in power of the two 
techniques is slight. 

One detail that has not been mentioned in this section 
is how to deal appropriately with ties in the data. If ties 
occur among the Z's when applying the first technique, 
simply use all permutations of the vector of midranks in the 
computation of the null distribution of T(0). The null 
distribution computed is the conditional null distribution 
of T(0), given the observed midranks, and the exact, dis- 
tribution-free properties of the test and interval are 
retained. We assume the Theil-Sen approach is only being 
applied when the sample size prohibits computation of the 
exact null distribution of T(0). In this case, modifica- 
tions of the Theil-Sen approach based on Kendall's tau in 
the presence of ties are referenced by Hollander and Wolfe 
(1973, p. 192), but they do not retain the exact nature of 
the test and confidence interval. 

As discussed in Section 2.4, the Hollander and Rao-Gore 
methods also provide exact, distribution-free tests of 
HqJ 3^_2 = and confidence intervals for the slope differ- 
ence, 3-j^_2. The null distributions required to use their 
methods have been tabulated and are readily available for 
several sample sizes. Hence these methods are competitors 
to the exact, small sample techniques proposed in this 
section. Monte Carlo comparisons of the power of these 
tests are given in the next section. 



75 



2.6 Monte Carlo Results 
To compare the powers of the test statistics discussed 
in the previous sections, we conducted a Monte Carlo study. 
The study concentrated on unequally spaced regression 
constants for the following reasons. The asymptotic rela- 
tive efficiency results in Section 2.3 indicate superiority 
of the Sievers-Scholz procedure in a wide variety of cases 
when the regression constants are equally spaced. Since the 
structure of the Sievers-Scholz statistic T (b) (2.1.5) 
utilizes information about the spacing of the regression 
constants, one would expect that its relative performance 
improves when unequal spacing is used. Hence, we were 
particularly interested in comparisons of the test statis- 
tics under unequal spacing of the regression constants. We 
begin our discussion with a description of the sample sizes, 
regression constants, error distributions, and parameter 
values used in our simulation study. 

Choice of Regression Constants 

Recall from Section 2.4 that Hollander's technique 
requires a scheme for pairing the regression constants on 
each line to form slope estimates. Although Hollander 
clearly specifies the pairing scheme (2.4.3) under equal 
spacing (2.4.2) of the regression constants, there exists 
some ambiguity in the choice of such a scheme when the 
regression constants are unequally spaced. To avoid this 



76 



ambiguity, we use what we call mirrored spacing, which we 
now describe. 

Consider a smooth nondecreasing function t(-) defined 
over the interval [0,1/2]. Suppose t{.) maps this interval 
onto itself with t(0) =0 and tCl/2) = 1/2. Then we define 
2k regression constants, x^, x^, ..., x^j^, over the interval 
[0,1] by the relations 



"m = ^(k?l) ' 



^m+k = ^^m + 1/2 ' 

for m = 1, ..., k. Multiplication by a scale factor can be 
used to make these constants fall over a more natural range. 
We use the term mirrored spacing since, by definition, 

'm+k ~ ■^m "^ "'■/^ for ^ = 1/ , k, and so the arrangement of 

the set of x's {x^^^^, ..., x^^^} over [%,!] is identical to 
the arrangement of the set {x^, ..., x^^} over [0,^]. when 

constructing Hollander's statistic we pair responses at x 

m 

with those at x^^^, thus conforming to Hollander's recom- 
mendation to choose a pairing scheme approximating that used 
with equal spacing. Mirrored spacing also allows the use of 
the Rao-Gore statistic, which is not generally applicable 
with unequally spaced regression constants. 

We experimented with t(.) of the form t (u) = au^, where 
a is chosen to satisfy t(l/2) = 1/2 and i is a positive 
integer. Note that i = 1 results in two groups of equally 



X 



77 



spaced constants over the intervals (0,1/2) and (1/2,1). As 
i increases, the regression constants tend to group closely 
just below 1/2 and 1. For our simulation study we selected 
1-3, t(u) = (1/3) 'u . This choice of i results in regres- 
sion constants that depart sufficiently from equal spacing 
without the excessive clumping observed under larger values 
of i. 

Selection of Err or Distributions and Parameter Values 

To generate simulated random variates we used the 
Fortran subroutines of the International Mathematical and 
Statistical Library (IMSL) . We selected four error dis- 
tributions: uniform, normal, double exponential, and 
Cauchy. In terms of the heaviness of tails, these dis- 
tributions cover a broad range. They are listed above in 
order of increasing heaviness of tails, from the uniform 
distribution which has very light tails to the Cauchy 
distribution which has very heavy tails. The standard 
normal distribution (N(0,1)) was used, and scale factors of 
the other distributions were selected such that the proba- 
bility between -1 and 1 was the same for all four distribu- 
tions . 

The values of &^_^ at which the power was estimated 
were determined by selecting multiples of the estimated 
standard deviation of the difference of the least squares 
estimates of g^ and e^ in each case. These multiples were 
chosen to achieve a wide range of power. 



78 



Designs and Computational Details 

There are basically two parts to the Monte Carlo study 
presented here. In the first part we used moderately large 
samples and applied approximate tests based on asymptotic 
theory. The second part of the simulation used small sample 
sizes and exact nonparametric tests. The form of the 
classical t-test was the same in both parts of the Monte 
Carlo simulation. 

For the first part of the Monte Carlo study, dealing 
with moderately large samples, we chose two designs which we 
call design A and design B. In presenting the sample sizes 
used in each design, we give the number of regression 
■constants per line. Since we are assuming two lines, the 
total sample size is twice of what we give below. Design A 
consists of 3 replicates at each of 20 distinct regression 
constants, resulting in 60 observations per line. The 
20 distinct regression constants were selected by using 
mirrored spacing as described and multiplying by a scale 
factor of 20. The resulting regression constants are listed 
in Table 3. 

Design B uses 30 distinct regression constants, with 
one observation per line taken at each of these constants. 
The constants for design B were selected by generating 
30 random numbers between zero and one, and multiplying them 
by a scale factor of 30. We wanted one design chosen to 
allow power comparisons of the tests under regression 
constants whose spacing followed no structured pattern, and 



79 



Table 3. Regression Constants Used for Designs A and 



B, 



Design A 



Design B 



X. 



X, 



X, 



X- 



^10 



X, 



X, 



X, 



X,. 



X, 



X. 



Xr 



X, 



X 



X 



X 



'10 

11 

12 
'l3 

14 
15 



= 0.0075 
= 0.0601 
=. 0.2029 
= 0.4808 
= 0.9391 
= 1.6228 
= 2.5770 
= 3.8467 
= 5.4771 
= 7.5131 

= 0.65 
= 1.16 
= 5.46 
= 6.09 
= 8.34 
= 8.52 
= 9.50 
= 10.83 
- 13.61 
= 13.84 
= 13.93 
= 15.20 
= 15.42 
= 15.58 
= 16.51 



X 



X 



X 



X 



X 



X 



11 

'12 
13 
14 
15 
16 
17 
18 
19 



X 



20 



X 



X 



X 



X 



X 



X 



X 



16 
17 
18 
19 
20 
21 
22 
23 



X 



X 



X 



'24 
'25 
26 
27 
28 
'29 
'30 



= 10.0075 
= 10.0601 
= 10.2029 
= 10.4808 
= 10.9391 
= 11.6228 
= 12.5770 
= 13.8467 
= 15.4771 
= 17.5131 

= 17.59 
= 18.81 
= 18.83 
= 19.53 
= 19.99 
= 20.13 
= 20.78 
= 21.01 
= 22.76 
= 23.78 
= 25.86 
= 26.37 
= 26.59 
= 28.06 
= 28.92 



1) For Design A, 3 responses per line were observed at 
each of the 20 regression constants. 

2) For Design B, 1 response per line was observed at each 
of the 30 regression constants. 



80 



hence this accounts for our departure from mirrored spacing 
in this case. The regression constants for design B are 
listed in Table 3. To compute Hollander's statistic under 
design B, we used the usual grouping scheme, pairing the 
response at x^ with that at ^^+^5' ^^^^^ m = , 1, ..., 15. 
Note that given the arbitrary spacing of the regression 
constants, the Rao-Gore statistic is not applicable for 
design B. 

All tests applied in the simulations under design A and 
B used a nominal level of a = .05, and in each case the null 
hypothesis H^ : ^-^-2 ~ ° ^^^ being tested against the one- 
sided alternative H^ : ^^_^ > 0. The tests used were the 
Sievers-Scholz test described in Section 2.3, the Theil-Sen 
test (based on T* (b) in (2.1.2) with a = sgn (x -x ) ) , the 
Hollander test based on the statistic W defined in 
Section 2.4, the Rao-Gore test (design A) based on U defined 
in Section 2.4, and the t-test based on the difference of 
the least square estimates of 6, and g-- Each of these 
tests were employed at an approximate a = .05 level 
utilizing their respective asymptotic distributions. 

For comparisons of the various procedures under small 
samples we selected three designs, which we call design C, 
design D, and design E. All three designs consist of one 
response per line at each of the regression constants. 
Sample sizes per line of 6 , 8, and 12 were used in 
designs C, D, and E, respectively. Mirrored spacing was 



81 



used for all three designs. The design points were multi- 
plied by scale factors of 10, 10, and 15 for designs C, D, 
and E, respectively. The resulting regression constants for 
these three designs are given in Table 4 . 

When using designs C and D, the exact tests of 
^0" ^1-2 ^ ° against H^ : &^_^ > Q associated with each of 
the four nonparametric procedures were used. The exact 
Sievers-Scholz and Theil-Sen tests as discussed in Sec- 
tion 2.5 were applied. Thus the exact distribution of the 
Sievers-Scholz statistic T{0) was first computed for designs C 
and D to determine appropriate critical values. 
An example of a portion of this distribution for design C is 
given in Table 5. Exact versions of the Hollander and 
Rao-Gore tests rely on the exact null distributions of the 
Wilcoxon signed rank and Wilcoxon-Mann-Whitney statistics, 
respectively, as tabulated and discussed in Hollander and 
Wolfe (1973) . Randomization was used to bring these exact 
procedures to the same level. The natural a-levels of the 
tests were compared to select a level for each design at 
which the amount of randomization needed was minimal. A 
nominal level of a = .125 was used for design C while 
a = .057 was selected for design D. 

With 12 observations per line, design E does not allow 
the feasibility of the exact Sievers-Scholz test. Instead, 
we replaced this with an approximate procedure, randomly 
selecting 10,000 permutations of the rank vector and com- 
puting the proportion of these perm.utations resulting in a 



"-iM-^ti^*;, ■ |HMiH|ni(^i9i« 



82 



Table 4. Regression Constants Used for Designs C, D, 

and E. 



Design C ^i = 0.078 x^ = 5.078 

x^ = 5.625 

Xg = 7.109 

Design D x^ = 0.040 x^ = 5.040 

Xg = 5.320 

x^ = 6.08 

Xg = 7.560 



Design E x^ = 0.022 x^ = 7.522 

Xg = 7.675 
Xg = 8.090 
x^Q = 8.899 
x^^ = 10.233 
x^2 = 12.223 



^1 


= 


0.078 


^? 


= 


0.625 


^3 


= 


2.109 


^] 


= 


0.040 


^2 


= 


0.320 


^3 


= 


1.080 


^4 


= 


2.560 


^1 


= 


.022 


^7 


= 


.175 


^3 


= 


.590 


^4 


= 1 


.399 


^5 


= 2 


.733 


^6 


= 4 


.723 



For Design C, D, and E, one response per line was observed 
at each design point. 



^*=«!*«¥Va3»¥^-^-^i|||)l . 



83 * 



Table 5 . Upper Portion of the Exact Null Distribution of 
T(0) when using Design C. 



t(0) 



Pq {T{0) > t(0)} 



6.64062 
6.69269 
6.82289 
6.82292 

6.87499 
7.00519 
7.13542 
7.18749 
7.31769 
7.36979 
7.49999 
7.68229 
7.81249 
7.86459 
7.99479 
8.17709 



0.0583 
0.0542 
0.0528 
0.0514 
0.0486 
0.0472 
0.0444 
0.0431 
0.0389 
0.0347 
0.0333 
0.0236 
0.0194 
0.0167 
0.0139 
0.0111 



-^i-t*ut,rwi»(i«w«<**»rif--»"!**'*^'F— - 



84 



value of T(0) larger than the one calculated from the 
observed data. Rejection of H^ : Q^_^ = o in favor of 
^1' ^1-2 ^ ° occurred when this proportion was less than or 
equal to the nominal level, which was set at a = .047. 
Recall that this type of approximate procedure was discussed 
in Section 2.5. The implementation of this approximate 
procedure was facilitated by an algorithm due to Knuth 
(1973) , which presents a one-to-one association between the 
integers 1,...,N! and the N! permutations of (1,...,N). 
Because of the cost of this approximate Sievers-Scholz 
procedure, the number of simulations for design E was set at 
500, and only two error distributions were selected (normal 
and Cauchy) . 

Discussion of Results 

Empirical levels and powers of the Sievers-Scholz, 
Theil-Sen, Hollander, Rao-Gore, and classical tests under 
design A are presented in Tables 6, 7, 8, and 9, for the 
uniform, normal, double exponential, and Cauchy distribu- 
tions, respectively. Empirical levels vary but generally 
remain within two standard errors of the nominal .05 level. 
It is seen that in terms of their power, the Sievers-Scholz 
and Theil-Sen tests uniformly dominate the Rao-Gore and 
Hollander tests. The power of the classical t-test is 
highest under uniform and normal errors, but falls below the 
powers of Sievers-Scholz and the Theil-Sen tests for double 
exponential errors. The t-test has the lowest power of the 



85 



Table 6. Empirical Power Tiir.es 1000 Under Design A for the 
Uniform Distribution (a=.05). 



h-2 





.024 


.064 


.096 


Sievers-Scholz 


058 


192 


675 


934 


Theil-Sen 


059 


2 


- 18 


14 


Hollander 


063 


4 


- 57 


- 35 


Rao-Gore 


061 


- 15 


- 58 


- 36 


Classical 


061 


17 


71 


31 



The first row and first column give the empirical power 
times 1000. Entries in the rem.ainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



Table 7. Empirical Power Times 1000 Under Design A for the 
Normal Distribution {a=.05) . 




Sievers-Scholz 041 217 561 861 

Theil-Sen 036 5 - 21 - 15 

Hollander 046 - 10 - 70 - 51 

Rao-Gore 041 - 25 - 41 - 53 

Classical 042 33 27 26 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicatira a lower 
power than Sievers-Scholz. 



86 



Table 8. Empirical Power Times 1000 Under Design A for the 
Double Exponential Distribution (a=.05) . 



^1-2 





.032 


.064 


.096 


Sievers-Scholz 


052 


234 


476 


808 


Theil-Sen 


043 


- 16 


6 


- 14 


Hollander 


057 


- 46 


- 69 


- 116 


Rao-Gore 


055 


- 29 


- 53 


- 67 


Classical 


049 


- 16 


- 32 


- 22 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



Table 9. Empirical Power Times 1000 Under Design A for the 
Cauchy Distribution (a=.05) . 



^1-2 





.064 


.096 


.160 


Sievers-Scholz 


052 


394 


590 


896 


Theil-Sen 


052 


- 13 


1 


1 


Hollander 


053 


- 140 


- 245 


- 275 


Rao-Gore 


062 


- 41 


- 44 


- 31 


Classical 


043 


- 289 


- 451 


- 667 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



87 



five tests under the heavily tailed Cauchy distribution. 
The Sievers-Schclz test tends to have higher power than the 
Theil-Sen test, but the difference is not pronounced. 
Comparing Hollander's test with that of Rao and Gore, we see 
that the power of the Rao-Gore test dominates the power of 
Hollander's test for the two heavily tailed distributions 
(double exponential and Cauchy) but they perform about 
equally under normal and uniform errors. A summary of all 
the Monte Carlo results will be given at the conclusion of 
this section. 

Tables 10, 11, 12, and 13 present empirical levels and 
powers of the Sievers-Scholz , Theil-Sen, Hollander, and 
classical tests under design B for the uniform, normal, 
double exponential, and Cauchy distributions, respectively. 
Empirical levels of the tests appear somewhat depressed, but 
generally fall within two standard errors of the nominal 
.05 level. For the uniform, normal, and double exponential 
distributions, the order of the tests in decreasing power is 
classical, Sievers-Scholz, Theil-Sen, and Hollander. The 
dominance of the classical approach lessens with increasing 
heaviness of tails of the ■ error distribution. For Cauchy 
errors the classical test again falls into last place, with 
the Sievers-Scholz and the Theil-Sen tests exhibiting 
highest powers. 

For design C, Tables 14, 15, 16, and 17 present empiri- 
cal levels and powers under uniform, normal, double 
exponential, and Cauchy distributions, respectively. 



Table 10 



88 



Empirical Power Times 1000 Under Design B for the 
Uniform Distribution (a=.05). 













^1-2 





.034 


.068 


.102 


Sievers-Scholz 
Theil-Sen 
Hollander 
Classical 


039 
036 
037 
043 


253 

- 16 

- 62 

37 


651 

- 18 

- 161 

76 


934 

- 12 

- 89 
31 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



Table 11. Empirical Power Times 1000 Under Design B for the 
Normal Distribution (a=.05). 




Sievers-Scholz 
Theil-Sen 
Hollander 
Classical 



042 
038 
055 
046 



209 
20 
24 
25 



569 

- 28 

- 128 

48 



878 
18 

154 
26 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



^•"•^^rv* •■•«J'f ■"** c --'I'-* ■»— ^ J- 



89 



Table 12. Empirical Power Times 1000 Under Design B for the 
Double Exponential Distribution (a=.05). 




Sievers-Scholz 
Theil-Sen 
Hollander 
Classical 


047 
039 
045 
053 


201 

4 

- 21 

18 


495 

- 24 

- 137 

2 


753 

- 23 

- 171 





The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, v/ith a negative value indicating a lower 
power than Sievers-Scholz. 



Table 13. Empirical Power Times 1000 Under Design B for the 
Cauchy Distribution (a=.05). 




Sievers-Scholz 


044 


225 


577 


770 


Theil-Sen 


047 


6 


5 


7 


Hollander 


041 


- 90 


- 250 


- 324 


Classical 


043 


- 133 


- 377 


- 455 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



•mtfimmmmmspm*'^ 



90 



Table 14. Empirical Power Times 1000 Under Design C for the 
Uniform Distribution {a=.125). 




Sievers-Scholz 121 424 824 977 

Theil-Sen 125 - 27 - 61 - 37 

Hollander 113 - 53 - 101 - 50 

Rao-Gore 122 - 26 - 57 - 7 

Classical 128 55 51 17 



The first row and first column give the emcpirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



Table 15. Empirical Power Times 1000 Under Design C for the 
Normal Distribution (a=.125). 



^1-2 





.218 


.436 


.654 


Sievers-Scholz 


112 


371 


687 


918 


Theil-Sen 


117 


- 17 


- 48 


- 35 


Hollander 


110 


- 52 


- 88 


- 91 


Rao-Gore 


103 


- 30 


- 49 


- 44 


Classical 


117 


6 


58 


36 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a 'lower 
power than Sievers-Scholz. 



91 



Table 16. Empirical Pcv/er Times 1000 Under Design C for the 
Double Exponential Distribution (a=.125). 



^1-2 





.218 


.436 


.654 


Sievers-Scholz 


110 


354 


612 


854 


Theil-Sen 


117 


- 16 


- 20 


47 


Hollander 


123 


- 46 


- 41 


- 65 


Rao-Gore 


113 


- 25 


- 26 


34 


Classical 


111 


23 


61 


37 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz . 



Table 17. Empirical Power Times 1000 Under Design C .f or the 
Cauchy Distribution (a=.125). 



Sievers-Scholz 


131 


320 


465 


574 


Theil-Sen 


122 


- 13 


9 


1 


Hollander 


132 


- 57 


- 50 


- 41 


Rao-Gore 


125 


- 42 


- 25 


- 11 


Classical 


141 


- 23 


2 


4 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



'-n^TP'lV ^'/f-r^ t^*K-j* ^fr w^-; 



92 



Tables 18, 19, 20, and 21 present these same results for 
design D. Recall that designs C and D consisted of samples 
of size 6 and 8 per line, respectively, and the exact 
nonparametric tests were used under these designs. The 
relative performance of the tests using these small sample 
sizes is similar to that of the approximate tests under 
larger samples, except that the dominance of the classical 
t-test is more dramatic under the smaller sample sizes. 
Only for Cauchy errors under design D does the power of the 
Sievers-Scholz test clearly dominate the classical test. 
The Sievers-Scholz and Theil-Sen tests generally exhibited 
greater powers than the Hollander and Rao-Gore tests. The 
one exception to this occurred under uniform errors, where 
the observed power of the Rao-Gore test was slightly higher 
than that of the Theil-Sen test, although we note that in 
these cases the Sievers-Scholz test had highest power among 
these three tests. 

For design E, Tables 22 and 23 present empirical levels 
and powers for the normal and Cauchy distributions, respec- 
tively. When compared with the Theil-Sen test, the 
approximate Sievers-Scholz test had higher power under 
normal errors and lower power under Cauchy errors. Hence 
these results do not indicate a clear choice between these 
two methods. However, the Sievers-Scholz and Theil-Sen 
tests once again exhibit greater powers than the Hollander 
and Rao-Gore tests (except in one instance v/here the power 
of the Rao-Gore test marginally exceeded that of the 



1 »:;»J Tl C i»i« Tll, ■I JM"'i ifrT°" 



93 



Table 18. Empirical Power Times 1000 Under Design D for the 
Uniform Distribution (c=.057). 




Sievers-Scholz 


059 


300 


661 


Theil-Sen 


064 


- 26 


- 36 


Hollander 


056 


- 68 


- 160 


Rao-Gore 


064 


- 16 


- 25 


Classical 


057 


31 


81 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, V7ith a negative value indicating a lower 
power than Sievers-Scholz. 



Table 19. Empirical Power Times 1000 Under Design D for the 
Normal Distribution (a=.057). 











^1-2 





.186 


.372 


Sievers-Scholz 

Theil-Sen 

Hollander 

Rao-Gore 

Classical 


059 
056 
050 
054 
058 


223 
9 

- 41 

- 28 
34 


551 

- 41 

- 120 

- 47 
94 



The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



=»(i««jicsc;;i— ST*Mfii 



94 



Table 20. Empirical Power Times 1000 Unc3er Design D for the 
Double Exponential Distribution (a=.057). 



1-2 



Sievers-Scholz 

Theil-Sen 

Hollander 

Rao-Gore 

Classical 






.372 


.558 




059 


451 


684 




054 


- 11 


- 23 




053 


- 92 


- 135 




059 


- 22 


- 29 




063 


60 


90 





The first row and first column give the em.pirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



Table 21. Empirical Power Times 1000 Under Design D for the 
Cauchy Distribution {a=.057) . 



^1-2 





.372 


.930 




Sievers-Scholz 

Theil-Sen 

Hollander 

Rao-Gore 

Classical 


058 
057 
064 
057 
062 


342 

- 17 

- 71 

- 28 

- 65 


645 
17 

- 112 

- 63 

- 44 





The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding Sievers- 
Scholz result, with a negative value indicating a lower 
power than Sievers-Scholz. 



g i Q Pi gaog i. s^j!yit*giai 



95 



Table 22. Empirical Power Times 1000 Under Design E for the 
Normal Distribution (a=.047). 



'1-2 



100 



200 



Sievers-Scholz 
Theil-Sen 
Hollander 
Rao -Go re 
Classical 



058 


228 


576 


064 


- 18 


- 48 


056 


- 46 


- 98 


058 


- 46 


- 46 


062 


- 14 


46 



1) The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding 
Sievers-Scholz result, v;ith a negative value indicating a 
lower power than Sievers-Scholz. 

2) All powers based on 500 simulations. 

3) An approximation to the exact Sievers-Scholz technique 
was used by selecting a random sample of 10,000 permuta- 
tions of the rank vector. 



Table 23. Empirical Power Times 1000 Under Design E for the 
Cauchy Distribution (a=.047) . 



1-2 



.100 



.200 



Sievers-Scholz 
-Theil-Sen 
Hollander 
Rao-Gore 
Classical 



060 


146 


288 


062 


2 


36 


084 


- 34 


- 72 


046 


- 12 


- 18 


054 


- 32 


- 96 



1) The first row and first column give the empirical power 
times 1000. Entries in the remainder of the table are 
expressed as the difference from the corresponding 
Sievers-Scholz result, with a negative value indicating a 
lower power than Sievers-Scholz. 

2) All powers based on 500 simulations. 

3) An approximation to the exact Sievers-Scholz technique 
was used by selecting a random sample of 10,000 permuta- 
tions of the rank vector. 



96 



Theil-Sen test) . Again the classical t-test performs poorly 
under Cauchy errors and generally best under normal errors, 
although at 6-,_2 = .100 under normal errors, the approximate 
Sievers-Scholz test had the highest pov/er. 

Summary of Conclusions 

The Monte Carlo results presented here generally concur 
with the asymptotic relative efficiencies (AREs) discussed 
in Section 2.4. Hence we draw the following conclusions 
about the power of the tests compared in the two line 
setting assuming common regression constants. First con- 
sider comparisons of the Sievers-Scholz, Hollander, and 
Rao-Gore methods, all three of which include exact confi- 
dence intervals for the slope difference g, „. The AREs and 
Monte Carlo results indicate superiority of the Sievers- 
Scholz approach when compared with the other two exact 
nonparametric procedures. Thus we recommend the use of the 
proposed application of the Sievers-Scholz approach over the 
Hollander and Rao-Gore methods. 

Mow consider comparisons of the Sievers-Scholz test. 
Sen's test, and the classical least squares theory t-test. 
As rank tests both the Sievers-Scholz and Sen test can be 
expected to be robust to gross errors, an advantage these 
two tests hold over the classical t-test. Although Sen's 
test has higher ARE than the Sievers-Scholz test under 
heavily tailed distributions, recall that Sen's test 



— ■T-.T ^v»> J . , ■-. ,i,, n y,— ,— >^<»^,^.,n^^Trs^-7-r**>— -gsr ,i , ii>, P ->. Tn .i m -W, »iA..- »; '>f^»wv*'-<«V 



97 



is distribution-free only asymptotically and does not 
include exact confidence intervals for the slope difference, 
^1-2* ^^ addition, Monte Carlo studies by Smit (1979) and 
Lo, Simkin, and Worthley (1978) indicate the power of Sen's 
test under small sample sizes is quite conservative with 
respect to the corresponding classical least squares test. 
Hence the Sievers-Scholz method is preferred over Sen's 
technique when an exact, distribution-free test is desired 
that includes associated exact confidence intervals for the 
slope difference, &^_2- Although the classical test has the 
highest overall power when the underlying errors are normal, 
choice of the Sievers-Scholz test is appropriate when an 
exact test is desired that maintains good power over a range 
of distributions, including heavily tailed ones such as the 
Cauchy distribution. 

Our Monte Carlo comparisons show the Sievers-Scholz 
approach generally has greater power than the Theil-Sen 
technique when the regression constants are unequally 
spaced. When the approximate versions of these two tests 
are being used under large samples, the computations 
required by each are about the same. Hence in this case we ■ 
recommend the Sievers-Scholz technique over the Theil-Sen 
approach. When exact versions of the tests are being 
applied under sample sizes small enough to allow computation 
of the exact null distribution of the Sievers-Scholz 
statistic (8 or fewer observations per line) , we again 



98 



recommend the Sievers-Scholz technique over the Theil-Sen 
approach. For larger sample sizes when the exact Sievers- 
Scholz test is not feasible, we recomm.end the Theil-Sen test 
if an exact distribution-free test is desired. The portion 
of our Monte Carlo study that included the Sievers-Scholz 
approximation based on random samples of permutations has 
limited scope and produced mixed results. Thus, further 
investigation of this technique is required before a judg- 
ment of its usefulness can be made. 



CHAPTER THREE 
COMPARING THE SLOPES OF SEVERAL LINES 

3.1 Introduction 
In this chapter we consider the case of several regres- 
sion lines, assuming coimnon regression constants for all 
lines. We examine the situation where one of the lines is a 
standard, or control, with which all other lines are com- 
pared. Suppose there are k lines and assume, without loss 
of generality, that the kth line is the standard. The 
specific topic of this chapter is the comparison of the 
slopes of the first k-1 lines with the slope of the kth 
line. 

As an example of the situation we are considering, 
suppose a dentist is examining some measure of strength for 
several different experimental amalgams used to fill cavi- 
ties in teeth. He may wish to compare the strength of each 
of these amalgams with the filling material he has commonly 
used in the past. His standard, then, is the commonly used 
material. Suppose he measures the strength of samples of 
each material, including the standard, after soaking in 
water for 2, 3, 4, and 5 days. The strength of an amalgam 
will generally decrease with increased soaking time. For 
each amalgam., consider a simple linear regression of the 



99 



100 



response, strength, on the independent variable, soaking 
time. The slope of each regression line is a measure of the 
rate at which the amalgam's strength decreases v/ith 
increased soaking time. The methods suggested in this 
chapter apply when the dentist wishes to compare the slope 
of the line resulting from each experimental amalgam with 
the slope of the line resulting from the standard filling 
material. 

In the rest of this section we formally establish the 
linear model assumed and the null and alternative hypotheses 
under consideration. We propose a statistic applicable in 
this setting whose form follows as a generalization of the 
statistic T(b) (2.1.5), discussed in Chapter Two. In 
Section 3.2 we derive asym.ptotic distributions of the 
proposed statistic, and specify a test of the null hypo- 
thesis based on the statistic. In Section 3.3 we illustrate 
how our proposed statistic can be used to conduct exact 
tests of the null hypothesis. A competitor to the proposed 
test is defined and explained in Section 3.4. We close this 
chapter in Section 3,5 v/ith Pitman asymptotic relative 
efficiencies comparing the proposed test with its com.peti- 
tor . 

We now establish the linear model assumed throughout 
this chapter. Let 

Y . . = a, . + e .Y. . + E. . , f31M 



101 



where i = 1, 2, ..., k, j = 1, ..., N, and 

^2 — ^2 — ' * ' - ^N * "*"" this model a . and g., i = 1, ,.., k, 
are unknown regression parameters, the x's are known 
regression constants, and the E's are mutually independent, 
unobservable random errors. We wish to test the null 
hypothesis 

-0* ^1 " ^2 " ••• " ^k (3.1.2) 

against the alternative 

-1* ^i ^ ^k ^°^ ^^ least (3.1.3) 

one i, i = 1 , . . . , k-1 . 

Thus the null hypothesis (3.1.2) is that all k lines are 
parallel. Note that this hypothesis can be expressed in 
terms of the slope differences, B- -B,, i=l, ..., k-1: 



So= ^1 - ^k = ^2 - \ = ••• = ^k-l - ^k = ° 



Define 



? 



^i: = ^ij -^kj' 



i - 1, ..., k-1, j - 1, ..., N. In terms of the underlying 
model parameters. 



102 



^ij = ("i-°'k) + (^i-^k^^j + (^ij-\j)' (3.1.4) 

and so for fixed i, the differences Z. ., i = i n 
follow a simple (single line) linear regression model. The 
case k = 2 (i=l) was the topic of Chapter Two of this 
dissertation. Recall in that case our proposed statistic, 
T(b) (2.1.5), results from the use of a statistic originally 
suggested by Sievers (1978) and Scholz (1977) for the single 
line setting, when applied to the differences Z. ., i = l 
..., N. Analogously, here we define for i = 1, ..., k-1 

T. (b.) = (1/N) ZZ X sgn(Z. -Z . -b . x ), (3 15) 
X 1 ^^^ rs ^ 'is ir 1 rs ' k^.x.o) 

where b^ are constants, x^^ = x^ - x^, and sgn ( • ) is as used 
in (2.1.2). Note that T^ (b) is T(b) (2.1.5) of Chapter Two 
when only considering the ith and kth lines. Thus from the 
results in Section 2.3 and 2.5, T^(0) can be used to test 
^Oi' ^i " ^k ag^i^st H^^: 3j_ ¥^ &^. Let 



2 N - 2 
^x = .^ (^i"^) /N (3.1.6) 

j=l -J 



and 



4, A = N'^x/^' (3.1.7) 



as in Chapter Two. 



103 



By Theorem 2.2.1 it follows that under H 



Oi' 



'i ~ ^k ^"■'^ 



certain regularity conditions, the asymptotic (II^-) distri- 
bution of 



T.(0) 



T,A 



is standard normal. Note also that 



k-1 
Q^0i=20' 



(3.1.8) 



that is, Hq holds if and only if h^^ holds for all i = 1, 

..., k-1. The asymptotic* normality of each T.{0) under H 

X oi 

and the relationship C3.1.8) between the H^^ and H suggest 
a possible approach to constructing a statistic for testing 
Hq using the T^(b^), i = l, ..., k-1: 
Let 



T = 



-1 
^T,A 



T^{0) 
T2{0) 



Vl(0) 



(3.1.9) 



Suppose we can show that under H^ , the asymptotic distribu- 
tion of T is multivariate normal with mean vector (zeros) 
and nonsingular (k-1) by (k-1) variance-covariance m.atrix 



104 



i;.. Denote, the inverse of |; by | . Then it follows from 
results in Serf ling (1980, p. 25, 128) that the asymptotic 
(N-^°°) distribution of 



S = T ^"^ T (3.1.10) 



under H^ is central chi-squared with k-1 degrees of freedom. 
This asymptotic distribution could be used to construct an 
approximate level a test of H^ based on S. Our purpose here 
is to motivate the use of S (3.1.10), a suitable quadratic 
form in the elements of T, to test H . In the next section 
it will be shown that S does have an asymptotic chi-squared 
distribution under given conditions. The specific form of f 
and the test of H^ (3.1.2) against H^ (3.1.3) will be given, 
along with the distribution of S under a certain sequence of 
alternatives. 

3.2 Asymptotic Theory and a Proposed Test 
In this section we develop the necessary asymptotic 
theory to derive a test of H (3.1.2) versus H (3.1.3) 
based on a suitable quadratic form in the elements of T 
(3.1.9). We also give the asymptotic distribution of T 
under a certain sequence of alternatives. This distribution 
will be needed in Section 3.5, when deriving asymptotic 
relative efficiencies. 

We first give some notation that will be needed in this 
section and the rest of Chapter Three. Some notation from 



"l'"*»W^VU'»i»*,*:t^.-i.«» -I 



105 



i 



Chapter Two will be repeated here, for the convenience of 
the reader. 

Basic notation . Let ^ indicate convergence in proba- 
bility and _^ indicate convergence in distribution. Let 



2 N - 2 
a^ = Z (x.-x) /N, (3.2.1) 

j=l -" 



2 N - 2 2 
""t A " ^ (x.-x) /3 = Na;/3, (3.2,2) 

' i=l -^ 



s-1 
^"s " .^ ^^s"^j^ ' (3.2.3) 



N 
X • = Z (x.-x ) , and so (3.2.4) 

j=r+l -■ ^ 



x.j - Xj. = N(Xj-x) (3.2.5) 



by simple algebra. If the event A occurs with probability 
one we write A, w.p. 1. We adopt the 0(«), o(«), (•), 
o (•), and ^ notations as in Serfling (1980, p. 1, 8-9). 



Matrix notation . Some matrix notation v/ill be needed, 
A vector of size n is considered to be an n by 1 matrix (a 
column vector) . Two such vectors that will be needed are 



106 



X = 



(3.2.6) 



n 



a vector of constants used in forming linear combinations, 
and 



= 








(3.2.7) 



the vector of zeros. The dimensions will be implied by the 
context. Also let 



^k = 



1 








• • • 








1 





• • • 











1 


• • • 






... 1 



(3.2.8) 



the identity matrix of order k, and 



^k = 



1 1 1 .. . 1 
1 1 1 ... 1 



1 1 1 ... 1 



(3.2.9) 



107 



the k by k matrix whose elements are all one (1) . When 
multiplication of a matrix by a scalar is indicated, all 
elements of the m.atrix are multiplied by the scalar, i.e.. 



5J3 = 



5 5 5 
5 5 5 
5 5 5 



The inverse of a matrix, if it exists, is indicated by a 
superscript -1, i.e., I"""- | = I^^. Finally, X' indicates the 
transpose of the vector _A.' 

Distributional notation . We assxime the underlying 
error distribution of all lines is the same, with cdf F and 
pdf f. Let E . , j = 1, 2, ..., designate independent and 
identically distributed random variables with cdf F. G and 
g denote, respectively, the cdf and pdf of E - E . The cdf 
and pdf of E^ + E2 - E- are denoted by M and m, 
respectively. Let 



A(F) = P{E^<E2+E3-E^, E^<E2+Eg-E^} - 1/4 



(3.2.10) 



and note that A (F) may also be written 



A{F) = /[l-M(e)]''dF(e) - 1/4 



(3.2.11) 



if the integral exists. Let 



108 
A* = 12A(F), (3.2.12) 

and for general pdf q let 



Kq) = /q^(x)dx, (3.2.13) 



provided the integral exists. For the multivariate normal 
distribution of dimension k with mean vector u and 
covariance matrix ^ we use the notation 

N^^ii'l)- (3.2.14) 

For the chi-squared distribution with k degrees of freedom 
and noncentrality parameter 6 we use the notation 



2 
X (<S)- (3.2.15) 



Hypotheses . The following hypotheses will be referred 
to frequently: 

-0' ^1 " ^2 " ••• " ^k (3.2.16) 

-1' ^i ^ ^k ^°^ ^^ least one i, (3.2.17) 
i = 1, . . . , k-1 

^Oi= ^i = ^k (3.2.18) 

^li= ^i ^ ^k (3.2.19) 



109 



Hj,: B. = Bj^ + ((../(Nk)^a^) (3.2.20) 



For expectation under H^ we use E . 

Conditions. The following set of conditions will be 
used in the statements of the theorems in this section: 



^ - 2 2 
I. I (x.-x) = Na -> ~ as N ^ cc. 

j = l ^ ^ 



II. max (x.-x)^ max (x.-x)^ 

l^J^N ^ 1^<N ^ 

^iq^^ - — 2 ->• as N > 

2 (x.-x) 2 Na^ 
j=l ^ 



III. G is continuous. 



IV. G has a square integrable absolutely 
continuous density g. 



We now prove that the asymptotic distribution of T 
(3.1.9) is multivariate normal, as conjectured in the 
previous section. We first state and prove two lemmas. 

Lemma 3.2.1; Let {X^, n ^ 1 } be independent bounded 
random variables with 



X 



nl 1 ^n' ^'P* ^ for n ^ 1. (3.2.21) 



— - •!*— ^ ■ii»»,'>j1i»|t^— - ■^f^.. 



110 



^ 2 ^ 9 

Let S^ = IX. and s = I E (X -EX ) ^ n > 1, and assxme 
3=1 J " j=i J J 

2 
s^^ -> CO as n -^ CO. If the bounds B satisfy 



^n = °^^n^ (3.2.22) 



then 



S - ES 

i ^ . N(0,1). (3.2.23) 

n 



Proof of lemma ; 

^^^ ^n = ^n ~ ^^n- "^^^^ ^<^n^ = °' ^^^ 



^^ ^ j!l^^''j"'=''j^^= -Ei^^i^- (3.2.24) 



In view of Liapounov's corollary (Chow and Tiecher, 1978, 
p. 293) to the Lindeberg-Feller central limit theorem, the 
result (3.2,23) follows if we can show the following 
conditions hold: 

For some 6 > , 



1) E|Yj2+fi <„,„>! (3.2.25) 



2) ,^_^E|Yj| ^ = o^^n"^"^) as n -^ -. (3.2.26) 



Ill 



We will show (3.2.25) and (3.2.26) hold for 6=1. From 

(3.2.24) we see eIy P < E(2B )^ = 8B"^ < «, n > 1, and so 

n ' — n n — 

(3.2.25) holds for 6=1. Now 



z e|y.|^ i: e(|y |2|y I) 
j=l -' ^ 2^ 



So 



3 3 

n n 



Z^E(|Yj| 2B.) (^^^^ (3.2.21)) 



n 
2 max B-. Z E( |y. | ) 

n 



2 max B . 
s 



(from (3.2.24) ) . 



n 



E E I Y . I 2 max B . 
izl_i_ < _IlJin_2 , (3.2.27) 



If V7e can show 



■ umnii 'i L i ii i iL-j ii jiM«j i i. i M ii j i» . i m f tf « »j i n ■! Il l »i »ii«a »» ir 



112 



2 max B . 



n 



(3.2,28) 



as n ^ «, then the Liapounov condition (3.2.26) holds for 
5 = 1 and the proof is complete. In the following let m be 
an integer such that m + 1 < n: 



max B 



max B . max B . 
l^i^n ^ l:£i^m ^ _^ m+l^i^n ^ 

n n n 



max B . 
^ l^j^m -* , , , . 

1 s ^ ^^^ (B./s.). 

n m+l<j<n -' -^ 



(3.2.29) 



Hold m fixed and let n -> «> in (3.2.29) 



max B . 

lira i£^lIL_ < + sup (B./s.). 
n-*-" n j>m+l -' -^ 



(3.2.30) 



Since the left-hand side of (3.2.30) does not involve m, we 
let m -> »: 



max B . 

lim — !££ — = lim sup (B./s.) = 0, 

n->-<» n m>°° j>m+l -' ^ 



(3.2.31) 



where the right-hand side of (3.2.31) equals zero because 
Bj = o(Sj) (3.2.22). So (3-2.28) is true and the proof is 
complete. 



113 



Leirana 3.2.2 ; Define for i = 1, , k-1. 



N 



T. (0) 



,^ E [T (0)|E -E ], 
= 1 -0 ^ ^3 ^3 



(3.2.32) 



where T^(0) is T^(.b^) in (3.1.5) with b. set to zero. Let 



T = a 



-1 
T,A 



T^(0) 
T2(0) 



(3.2.33) 



L 



Tj,_,(0) 



Under Iq* ^1 " ^2 " * 
III, 



- 3j^, assuming conditions I, II, and 



i t \^i^9.4) 



(3.2.34) 



as N ->■ "^ where 



t = Vi + A* (\.r\-i^' 



(3.2.35) 



and A* is given by (3.2.12). 



114 



Proof of lemma : 

From (3.1.4) it follows that under H^ , 



Z. -Z. =E. -E, -E. +E, , 
IS ir IS ks ir kr 



Then 



E„ [T. (0) E. .-E, .] 
H^ 1 ' ij kj-" 



= E„ [4 SZx sgn(Z. -Z . ) Ie. .-E, .] 

H„'N rs ^ 'is ir' ' 11 ki-' 

—0 r<s J -I 



E[rz izx sgn(E. -E, -E. +E, ) Ie. .-E, .] 
N rs ^ 'is ks ir kr' ' ig k:]-" 



= ^^ZZx^3E[sgn(E.3-E^3-E.^+E^^) |E. ^-Ej^.] . (3.2.36) 



For notational simplicity let D = E. .- E, . . Then examining 

13 K^] 



E[sgn(E.^-E^^-E.^+Ej^P |e. j-E,^..D] , (3.2.37) 



note the following: 



1) If s = j then D is independent of E. - E, and so 
(3.2.37) equals 



115 



P{D>E, -E, } - P{D<E. -E, } 
ir kr xr kr"^ 



= G(D) - {l-G(D)) 



= 2G(D) - 1. (3.2.38) 



2) Similarly, if r = j, then (3.2.37) equals 



1 - 2G{D) . (3.2.39) 



3) If r 7^ j and s ?^ j , then D is independent of 
^is " ^ks " ^ir ■*■ ^kr ^^'^ ^° (3.2.37) equals 



E[sgn(E.^-Ej^^-E.^+Ej^p]. (3.2.40) 

Now E^^ - Ej^g - E^ + E, is symmetric about zero so 
(3.2.40) equals zero. 
Recall the notation x. (3.2.3) and x . (3.2.4). 
Applying 1), 2), and 3) above to (3.2.36) results in the 
following: 



%tT.(0)lE..-E3^.] 



= TT ZZx E[sgn(E. -E, -E . +E, ) Ie. .-E, .1 
N _^ rs ^ ' IS ks ir kr' ' 13 k^ 



116 



2 J-1 N 

n[ r X (2G(D)-1) + r x(l-2G(D))] 

r=l J s=j+l -"^ 



|[{2G(D)-l)x_ + (l-2G(D))x..] 



^(2G(D)-1) (x.^-x^.) 



= ^(2G(D)-l)N(x.-x) (from (3.2.5)) 



= (2G(D)-l)(x.-x). (3.2.41) 

Substitute (3.2.41) into (3.2.32): 

N 



T,(0) = fjE2^[Ti(0)|E..-E^.] 



N 
= T (2G(E -E )-l) (x.-x) 



N 

Z_^2(Xj-x)G(E^j-Ej^j). (3.2.42) 



We have now established the form of the T. (0), i = 1, 
k-1. Consider an arbitrary linear combination of the 
i\(0) -s. 



117 



k-1 . 

.f/i^i^O). (3.2.43) 



Our basic approach to proving this lemma is to show the 
asymptotic normality of this linear combination (3.2.43). 
Let 

k-1 
Uj = 2(x.-5)_Z A.G(E -Ej^n. (3.2.44) 



i=l 



From (3.2.42) , 



k-1 . k-1 N 

J^A,T,(0, - ,f^VJ^2(x.-5,G(E,.-E,., 



N _ k-1 

= Z [2 (x.-x) Z A.G(E. .-E, .) ] 
j=l 3 i^i 1 13 k:' J 



N 
■ l^^r (3.2.45) 



Note that U^ and U^ , are independent, j 7^ j ' . Before 
deriving the mean and variance of U , observe the following: 
1) The cdf of E^j - E^. is G (i<k) . Let g"^ be an 

inverse of G, i.e., G"^(G(t)) = t , <_ t ^ 1 . G~^ 

exists by continuity of G (condition III) . Then 
the cdf of G(E^j-E^.) is given by 



-.--.-.-.^^1.3= 



118 



P{G(E..-Ej^.)<t} 



Pi'E^^-E^.<C~^ it) } 



G(G ^(t)) 



= t, < t < 1. 



Thus G(E^j-Ej^. ) is uniformly distributed over 
[0,1] . It follows that 

E[G(E. .-Ej^.)] = 1/2, (3.2.46) 

and 



E[G(E. .-Ej^.)-l/2]2 = 1/12, (3.2.47) 



the variance of a uniform [0,1] random variable. 

2) Recalling the definition of A (F) (3.2.10), it 
follows that 



A(F) + - = P{E2<E^+E^-E2, E2<E3+Eg-E^} 



= E tP{E5-E4<E^-E2 ,E^-Eg<E3-E2 | E^=e^,E^=e^,E^=e^}] 



119 



= E[P{E5-E^<e^-e2,E^-E^<e3-e2|E^=e^,E2=e2,E3=e3}] 

PiE^-E^<e^-e^\E^=e^,E^=e^,E^=e^}] 

= E[G(E^-E2)G(E3-E2)] , 

and so using (3.2.46) we have 

E[{G(E^-E2)-l/2) (G(E3-E2)-l/2)] = A (F) . (3.2.48) 



Then 



k-1 
EU. = E[2(x.-x) Z A.G(E. .-E, .)] 



= 2(x.-5)_Z^A.E[G(E..-E3^.)] 



k-1 



(Xj-x)_z X^ (from (3.2.46)), (3.2.49) 



and 



120 



Var(U.) = E[U.-EU.] ^ 



k-1 ^ 



= E[(2(x-x) Z X G(E -E ))-((x,-x) I A.)] 
-» i=i -^ -^J ^-J J j__2 1 



k-1 
= E[2{x.-x) I X. (G(E.j-Ej^.)-l/2)]2 



- 2 '"-I 2 
= 4{x -X) { I X^E[G(E.,-E, .)-l/2] 



+ 2,2Z^A.X.,E[{G(E. .-Ej^.)-l/2) (G (E . , ,-Ej^ . ) -1/2) ] } 



- 2 ^-1 P 
= 4(x -X) [(1/12) E A^ + A(F)2 ZI A.A.,], (3.2.50) 

i=l i<i' ^ ^ 



where (3.2.47) and (3.2.48) were applied to arrive at the 
final expression (3.2.50). Now we show that the conditions 
of Lemma 3.2.1 apply to the U.. First note 

lu^l = I2(x^-5)_I^A.G(E.^-E^^)| 



< 2 max ix -x| z | A | , w.p. 1 
l<j<N -' i=i ^ 



121 



k-1 



Let B = 2 max |x -x| z |^ |, and then 

l<j<N -' i=l ^ 



U^l 1 B^, w.p. 1, N > 1, (3.2.51) 



SO (3.2.21) of Lemma 3.2.1 is satisfied. Applying (3.2.50), 

2 ^ 
if we let s = z Var(U.), then 

j=l ^ 



2 ^ - 2 ^-^7 
s^ = 4 S (X -x)^[(l/12) Z Af+A(F)2 ZZ A.A.,] 

j=l ^ i=l ^ i<i' 1 ^ 



N _ k-1 _ 
= (1/3) Z (x,-x)^[ Z Xf+2A* ZZ \.X.,]. (3 2 52) 

j=l ^ i=l ^ i<i- 1 1 U.^.i^) 



For later reference we note: 



4 = Varjj^(ZA.T.(0)), (3.2.53) 



and taking A. = 1, a^, =0, ± f x' the form of the null 
variance of T^(0) follows from (3.2.52): 



r.2 2 



Var_^_^(T.(0)) = (1/3) Z^(x_.-x)^ = a^^^. (3.2.54) 



122 



Now examine B„/s : 
N N 



k-1 
max |x.-x|2 Z \x. 
B^/Sm = l^i^N -J i=l ^ 



{(1/3) Z (X -x)'^[ Z X^+2A* EZ X.X.,]}^ 
j=l ^ i=l ^ i<i' 1 1 

- as N - " (3.2.55) 

by condition II, and so (3.2.22) of Lemma 3.2.1 holds. We 
note from (3.2.49) , 



N N k-1 
2 EU. = Z (x.-x) Z A. = 
j=l ^ j=l 3 ,i=i 1 



Applying Lemma 3.2.1, 



N N 

2 U . - Z EU. ^ 

1=1 ^ j=i : d 

-^ i ^-^^— ^ N(0,1) 



N 



as N ->■ » , 



N 

Z U. ^ 

-^i > N(0,1) (from (3.2.56)), 



(3.2.56) 



123 



:> 



k-1 . 

Z A.T. (0) 
i=l d 



=> 



N(0,1) (see(3.2.45) ) , 

N 



_1 k-1 . 
a Z XT. (0) 

i=l d 
^I ;N(0,1), (3.2.57) 



as N ->• " 



Let A be a vector as in (3.2.6) and recall the definition of 
the matrix | (3.2.35). From (3.2.52) the following 
expression results: 



,-2 2 , -2 



N 



- 2 



''t^A^N = f^T,A(^/3) Z (X -x)^]A'tX 

j=l -^ 



= ^^tVt,a^A'^A 



= A'iA. 



(3.2.58) 



Using (3.2.58) in (3.2.57) we see 

=%fN(0,l) (3.2.59) 



A'l d 



ix'tx) 



as N -^ CO, where T is given by (3.2.33). The result (3.2.59) 
essentially shows that any linear combination of the 



124 



elements of T converges in distribution to a random variable 
with the distribution of the same linear combination of a 
\_l(0.4) random variable. By a theorem in Serfling (1980, 
p. 18) this is equivalent to showing T has an asymptotic 
(N->") Nj^_;l^^'^^ distribution, and the proof of Lemma 3.2.2 
is complete. 

Theorem 3.2.3: Under H^ : 3^ = 33 = . . . = 6j^, assuming 
conditions I, li, and III, 



d 

"k-l^-' + '' (3.2.60) 



T " N,,_^(0,t) 



as N -. -, where T is given by (3.1.9) and t is given by 
(3.2.35). 

Proof ; 

Consider f or i = 1 , ..., k-1, 

E„ (T. (0)-T. (0))^, 
-0 ^ 

Where T^(0) is given by (3.2.32). T^ (0) is called the 

projection of T. (0) on the Z..=E..-E i=l n 

under H^ . By a lemma due to Hajek and appearing in Serfling 

(1980, p. 300), it follows from the form of T. (0) that 



E (T.(O)-T. (0))2 = var^ (T.(0)) -Var^ (T, (0)), 
u —0 —0 



(3.2.61) 



125 



1, ..., k-1. From (3.2.54), 



Var^^ (T^(0)) = aj^^, (3.2.62; 



and from Theorem 2.2.1 applied to this situation. 



Varjj (T^(0)) = a^^^ + o^. (3.2.63) 



Substituting (3.2,62) and (3.2.63) into (3.2.61) yields 



E (T.(0)-T.(0))2 = ia^^-^ol) - a^^ = o^ 



■■> lim a \e (T. (0)-T (0))2 = lim a^ o^ 



N - 2 

2 (Xj-X)^ 

j=l ^ 



= 0, (3.2.64) 



for i = 1 , — , k-1 . Then it follows that 



crT^^^T^(O) - T^(0)) I (3.2.65) 



as N ^ ", i = 1, ..., k-1. This implies 



126 



_l k-1 k-1 



^T^A^^^i^i^O) - ^ ^iTi(O)) P (3.2.66) 

1=1 i=i -^ -^ 

as N ^ oo. Using the matrix notation (3.2.6), (3.2.33), and 
(3.1.9), we can express (3.2.66) as follows: 



X'T - X'T P 



(3.2.67) 



as N ^ 00. Thus from Slutzky's theorem (Randies and Wolfe, 
1979, p. 424), (3.2.67) implies that if X' T has a limiting 
distribution then X'T has this same limiting distribution. 
In Lemma 3.2.2 we saw that under H^ , assuming conditions I, 
II, and III, 



X ' T 

~ ~ t: t N(0,l) 



ix'tx) 



as N -> 00 (from (3.2.59)), and so 



k . N(0,1) (3.2.68) 



iX'iX) 



as N ^ 00. As argued in Lemma 3.2.2, this is equivalent 
(Serf ling, 1980, p. 18) to showing T has an asymptotic (N->oo) 
N]^-l(£'t) distribution, and the proof of Theorem 3.2.3 is 
complete. 



127 



Notice that | (3.2.35) has the form 



abb 
b a b 
b b a 



b b b . 



b 
b 
b 



(3.2.69) 



with a = 1 and b = a* = 12A (F) . Then by Theorem 8.3.4 in 
Graybill (1969, p. 171), | has an inverse if and only if 
(iff) 



1 7^ 12A(F) , 



(3.2.70) 



and 



1 7^ -(k-2)12A(F) . 



(3.2.71) 



Mann and Pirie (1982) give the following bounds for A (F) : 



1/36 < A(F) < 1/24. 



(3.2.72) 



The bounds (3.2.72) hold for all continuous distributions. 
Condition (3.2.70) fails iff a (F) = 1/12, which, in view of 
(3.2.72), is not possible for continuous F. Condition 

(3.2.71) fails iff A(F) = -[12(k-2)]"^ and k ^ 3. By 

(3.2.72) we see that A (F) cannot be negative, and so 



128 



(3.2.71) holds for continuous F. Hence for continuous F, 
^ has an inverse, i~ . Again appealing to Graybill's (1969, 
p. 171) theorem, the inverse of | is given by 

t = riA^^^k-l ~ l-(k-2)A* "^k-l^ • (3.2.73) 



Consider the quadratic form 



S = T'f^T, (3.2.74) 



T as in (3.1.9). Using (3.2.73) and some algebra, S can be 
expressed: 



k-1 ^ k-1 
(l+(k-2)A*) I (T^(0))-A*( S T^(0))^ 

S= ^^ ^=^^ . (3.2.75) 

a^^^a-h*) (l+(k-2)A*) 



The following gives the asymptotic distribution of S under 

Theorem 3.2.4 : Under H^ : 3^ = g^ = . . . = g^, assuming 
conditions I, II, and III, 



S ^ X^_i (0) (3.2.76) 



as N -» «, where S is given by (3.2.74) 



129 



Proof ; 

From Theorem 3.2.3, 



1 J X 



as N -»- ", under H^ , where X is a random vector having the 
N]^_l(£4) distribution. Then from Corollary 1.7 in Serf ling 
(1980, p. 25), it follows that 

S = T' Z~"^T J X' t~'''X 

as N ^- ", under Hq . Applying a theorem in Serf ling (1980, 
p. 128) it follows that 

X'T'^X 

2 

has the Xj^_j^(0) distribution, and the proof is complete. 

The result of Theorem 3.2.4 suggests a test of 

-0* ^1 " ^2 " ••• " ^k ^^- -1 (3.2.17). Let X^_^ be the cdf 
of the central chi-squared distribution with k-1 degrees of 
freedom. For < a < 1, let ^^_^ ^ be the upper 100 (1-a) 
percentile, that is X(iJ;j^_^ ^) = i _ a. Then from Theorem 
3.2.4, the test based on S that rejects H_ (3.2.16) if 

^■^'^k-l,a (3.2.77) 



is an approximate level a test of H against H (3.2.17) 



130 



Of course the parameter A (F) depends on the underlying 
distribution and so the value of S depends on the underlying 
distribution. Consistent estimates of A (F) are discussed by 
Mann and Pirie (1982), however such estimates are often 
tedious to compute. Consider Table 24, which is adapted 
from Mann and Pirie (1982) and shows the value of A (F) to 
four decimal places for several common error distributions. 



Table 24. Values of A (F) . 



F Uniform Normal Logistic Exponential Cauchy Max. Min. 



A(F) .0409 .0402 .0398 .0394 .0379 .0417 .0228 



Source: Adapted from Mann and Pirie (1982) . 

The last two columns of Table 24 give the maximum and 

minimum values of A (F) for continuous F, discussed 

previously (3.2.72). We see the range of possible values 

for A(F) is about 0.02, and all of the values listed remain 

within 0.01 of the maximiom. The range of possible values 

for A* = 12A(F) is (l/2)-(l/3) = 1/6 or about 0.17 for 

continuous F (see (3.2.72)). In view of this small range we 

suggest a test based on S which replaces A* by a value A* 

c' 

chosen to yield a conservative test. We now indicate how to 

choose the value of A*: 

c 



131 



Let 



k-1 
T - I T,(0)/(k-l) (3.2.78) 

i=l ^ 



2 ^-^ - 2 
s^ = ^ (T. (0)-T)^/(k-l) , and (3.2.79) 

i=l 



2 2-2 
c^ = s^/ (T) . (3.2.80) 



Then using these definitions in the expression for S 
(3.2.75) we have 



2 

S = S(A*) = [(k-DT^/cT^ ] [-!t__ + 1 _ j^ 

1-A l+(k-2)A 

(3.2.81) 



Examining the derivative of S(A*) with respect to A*, 
S' (A*) , it can be shown that 

S' (A*) is >, = ,< 

(3.2.82) 

(k-2)^ - c^ 
as A* is >, = ,< — 



(k-2)Cy + (k-2)^ 



Let S(l/3), S(l/2) be the value of S(A*) when A* is set at 
it lower and upper limits of 1/3 and 1/2, respectively. 



132 



Let 



r 



1/3 if cl > 1"^ 



T 2 




A* =/ ^^^ " ""t , if kzi < ^2 ^ 4(k-2) 

"" \ (k-2)c^ + {k-2)'2 k^ - T - (3^^^ J 2 



1/2 if c^ < ^ . (3.2.83) 



Using (3.2.82) and the upper and lower bounds of A* it 
follows from some routine analysis that 



min S(A*) = S(A*). (3.2.84) 

(1/3) < A* £ (1/2) ^ . 



Then if we define S(A*) to be the value of S (3.2.75) using 

* 
A* = A , it follows from (3.2.84) that the test of H„ 

(3.2.16) which rejects in favor of H (3.2.17) when 



^(^^ ^ Vl,a (3.2.85) 



will be conservative. Its true level will be less than or 
equal to the nominal level a. However in view of the small 
range of A* under continuous F, the degree of conservatism 
of the test (3.2.85) should be slight. 

Suppose 0)^^, 0^2 ..., 0)^ are given constants such that 
ojj^ = 0, and consider the sequence of alternatives. 



133 



^N = 



'i = 3^^ + to^/{Nk) ^a^, 



(3.2.86) 



1 X y • * • f K I 



Let 



(0 = 



OJ- 



0). 



^k-1 



(3.2.87) 



The following theorems give the asymptotic distribution of T 
(3.1.9) and S (3.2.74) under H„. 

Theorem 3.2.5. Let ^, o), and T be given by (3.2.35), 
(3.2.87), and (3.1.9), respectively. Under H (3.2.86), 
assuming conditions I-IV, 



T t Nj^_i(2(3^)I(g)k-\,|) 



(3.2.88) 



as N ^ «>. 



Proof ; 

Let 0)^^ = oj^/N^^a^, i = 

of alternatives is H„: g. = 

— N 1 



1, ..., k. Then the sequence 
^k "*" "iN' ^ ^ ^' •••/ k. Let 



134 



^^-i^N) = ^t]a 






Vi (-"k-1 n) 



(3.2.89) 



As noted in Chapter Two, the limiting distribution of 

T = T{_0) under H^ is the same as the limiting distribution 



°f I(~%) under H 



. = 3, . So assume H- and 



"1 ^2 
show that T(-i;Jfj) has the desired limiting distribution. 

Consider an arbitrary linear combination of the T. (-u. ) 

1 iN ' 



k-1 

i=l ^ ^ iN' 



(3.2.90) 



Using the matrix notation (3.2.89) and (3.2.6), we can 
express the linear combination (3.2.90) as 



-t,aA't(-%) 



(3.2.91) 



Our basic approach to proving Theorem 3.2.5 is to show this 
linear combination has an asymptotic distribution that is 
univariate normal. We first state and prove two lemmas. 



Lemma 3.2.6 : Assume H 
conditions I-IV. Then 







^1 = 



= B, and 



135 



%f.!^^i^i(-^N^/^T,A^ - 2(3'^)I(g)k^^E^A.co. (3.2.92) 



as N ->- «> . 
Proof of lemma ; 

Applying Lemma 2.2.3, we have in the notation of this 
chapter. 



Ejj . fT^(-aj^j^)/a^^^] ^ 2 (3^) I (g) k^u^ (3.2.93) 



asN-><=°, i = l, ..., k-1, H_ . given by (3.2.18). Since H-^ 
implies Hq., i = 1, ..., k-1 (see (3.1.8)), the result 
(3.2.92) follows immediately. 

Lemma 3.2.7 ; Assume H- ; B- = Bj = ••• = S, and condi- 
tions I-IV. Then 



.^^i^i(-'^iN) - t.^^i^i^O) -^ V(.^^iTi(-'OiN))] 
1=1 1=1 —0 1=1 



''t,a 



(3.2.94) 



as N ->■ ". 



Proof of lemma; 



Applying Lemma 2.2.4, we have in the notation of this 
chapter. 



136 



^i^-^N^ - f^i^O)-^ V.(Ti (--,,))] 

Ol p ^ 

: (3.2.95) 

T,A 



asN^co, i = i^ ._^ k_l^ Since H. implies H„ . , 

—0 '^ 1 

i = 1, ..., k-1 (see (3.1.8)), the result (3.2.94) follows 
immediately. 

Proof of Theorem 3.2.5 (continued) ; 

Under H^^ and conditions I, II, and III, Theorem 3.2.3 
(see (3.2.68) ) showed 



X'T J N(0,X'|X) (3.2.96) 



as N -> CO, where ^ is given by (3.2.35). 

The result of lemma 3.2.6 in matrix notation is 



Ejj [A'I(-%)] ^ 2(3'2)I(g)A'iil (3.2.97) 



as N -> =0. Using (3.2.96), (3.2.97) and Slutzky's Theorem 
(Randies and Wolfe, 1979, p. 424), it follows that 



X'T + Ejj U'T(-co^)] J N(2(3'2)I(g)A'iil'i'lA) (3.2.98) 



as N ^ =°. The result of lemma 3.2.7 in matrix notation is 



^'T(-aij^.) - U'T + Eg (A'T(-aij^))] ^ (3.2.99) 



137 



as N ^ 00. So using (3.2.98) and (3.2.99), a second 

application of Slutzky's theorem shoxvs 



^'T(-a)jj) t N(2(3^)I(g)A'i!l/ A'ID (3.2.100) 



as N ^ ", under H^ . Put another way, (3.2.100) reveals that 
the asymptotic distribution of A'K-w^) under H is the 
distribution of ^'X, where X has the Nj^_^ (2 (3^) I (g) k\, ^) 
distribution. Hence by a theorem in Serfling (1980, p. 18) 
this is equivalent to showing T(-u ) converges in distribu- 
tion to this same multivariate normal distribution. Hence 
by our remarks at the beginning of the proof of Theorem 
3.2.5, the proof is complete. 

Theorem 3.2.8 ; Let S be given by (3.2.74). Under H 
(3.2.86), assuming conditions I-IV, 



k-1 , k-1 
(l+(k-2)A*) E a3T-A*( Z to . ) ^ 

S t xj_i([12l2(g)][ k(l-A*Ml+'(k-2)iM ^ ^) (3.2.101) 



as N -»■«>, A* given by (3.2.12) 
Proof: 



From Theorem 3.2.5, under H„, 

— N 



1 J X 



138 



as N -> CO, where X has the Mj^_^ (2 (3^) I (g) k^oj , t ) distribution, 
Then from Corollary 1.7 in Serfling (1980, p. 25), it 
follows that 



S = IT^T f X't ^X (3.2.102) 

as N > 00, under H^^. Applying a theorem in Serfling (1980, 
p. 128) , X' t~-^X has the 

Xj^_^(12l (g)k~^m'i~^w,) (3.2.103) 

distribution. The noncentrality parameter in (3.2.101) is 
just the one in (3.2.103) expressed in terms of the elements 
of 0) (3.2.87) and ^""^ (3.2.73). 

3.3 An Exact Test 

The statistic S (3.2.75) can be utilized to construct 
an exact test of H^ (3.2.16) against H^ (3.2.17). The idea 
is similar to that used for exact tests based on T(0), 
described in Chapter Two. After first establishing some 
notation, we will review the case of k=2 lines. We then 
extend this method to the situation of several lines. 

Recall the definition 



Z. . = Y. . - Y, . 

i: ID kj 



(a.-aj^) + (6-e^)x, + (E.j-E^j), (3.3.1) 



139 



where Y^^ follow the linear model (3.1.1), i = 1, ..., k-1, 

3=1, ..., N. Let R . . be the rank of Z . , amora 

(Z^^, ..., Zj^^}, the Z's resulting froir, the observations at 

the ith and kth lines. Consider the vectors 



R. 
-3 



R 



Ij 
Z 

2j 



R. 



k-1 j 



J 



(3.3.2) 



j - 1, ..., N, and the matrix composed of these vectors, 



R = [R^ R2 ... P^] . 



(3.3.3) 



The following representation of T. (0) in terms of the R^ 

1 i j ' 

Z 
•••/ Rj,j^T results from (2.5.1): 



^ 7 
T.(0) = (2/N) 2 Rf X . - (N+l)x 

j = l ^^ -' 



(3.3.4) 



When k=2, R has dimensions 1 by N. In this case 
-0 " ^01* ^1 " ^2 ^"^ under H^^ it follows from (3.3.1) 
that the Z^^, ..., z^^^^ are i.i.d. and hence the distribution 



140 



of R is uniform over the N! permutations of its elements, 
the integers 1, ..., N. Thus, as discussed in Chapter Two, 
the null (6^=62) distribution of T. (0) can be determined by 
using (3.3.4) to compute the N! equally likely values of 
T.(0). 

Now note that when k > 2, the matrix R is (k-1) by N 
and the elements within the jth column, R., are dependent, 

since their values all rely, in part, on the value of Y . . 

ik 

To take care of these dependencies, we condition on the 
values of the columns of R. That is, under H : 

^1 ^ ^2 "^ •*• ^ ^k' ^^^ distribution of R conditional on the 
observed values of R^ , j=l, ..., n, is uniform over the N! 
permutations of its columns. 

The conditional null distribution of R just described 
can be used to conduct an exact test of H against H using 
S(A^), defined after (3.2.84). The N! values of T (3.1.9) 
resulting from this conditional null distribution of R would 
be used to compute N! values of S(A*), each occurring under 
Hq v/ith conditional probability 1/N! . Suppose, for an 
observed set of data, s is one of these values such that 



P{S(A*)^s} = a. 



where the probability is evaluated using the conditional 
null distribution. Then, for this data set, the test which 
rejects H^ (3.2.16) in favor of H, (3.2.17) if 



141 



S(A ) ^ s 



is an exact, conditional level a test. Note that the degree 
of difficulty of computation of the given permutation 
distribution depends only on the number of regression 
constants per line, N, and not on the number of lines, k. 

o 

In view of this fact and the results discussed in 

Chapter Two, we found the exact test of H presented here to 

be computationally feasible with the use of a computer 
whenever N _< 8 . 

3.4 A Com.petitor to the Proposed Test 
The statistic S (3.2.74) resulted from an obvious 
generalization of the proposed application of the Sievers- 
Scholz statistic T(0) for use in comparing the slopes of 
two regression lines. Clearly one could consider such gen- 
eralizations based on the other statistics for testing 

— 0* ^1-2 ~ ^1 ~ ^2 "^ '^' P^ssented in Chapter Two. In view 
of the fare's reported in the previous chapter, we shall 
confine our attention to the statistic defined by Sen 
(1969) , keeping in mind that exact tests based on a permu- 
tation distribution are not possible for Sen's test 
statistic. We would anticipate that, in terms of PARE, 
Sen's test is likely to be better than the test based on S. 
Recall that the test based on S was designed to test 

lio= ^1 = ^2 = ... = 3, (3.4.1) 



142 



against 

-1* ^i "^ ^k ^^^ ^'^ least one i, f3.4.2) 

i = 1, . . . , k-1. 

Sen proposed his statistic for use in testing H against 
the more general alternative, 

^1 + ' ^ir •••/ 3j^ are not all equal, (3.4.3) 

and hence it would not be appropriate to compare his test 

against the test based on S. Instead we modify Sen's 

statistic to be specifically sensitive to the alternative H- 

(3.4.2). A test based on the modified statistic follows 

from the results of Sen. In the next section we compare 

this test with the test based on S . We note that the test 

based on Sen's results will not require common regression 

constants for all lines as required by the test based on S. 

However, as in Chapter Two, we will assume this is true to 

ease the notation and also to facilitate the comparisons in 

the next section. Thus we assume the basic linear model 

(3.1.1). The notation defined in Chapter Two when 

introducing Sen's statistic in the two line setting will be 

repeated here for convenience of the reader. 

Let (j) (u) be an absolutely continuous and nondecreasing 

function of u: < u < 1, and assume that <^ {u) is square 

integrable over (0,1). Let U,,. < U,„. < ... < U ,,,, be the 

(-L) (^) (N) 



143 



iii 



order statistics of a sample of size N from a uniform [0,1] 
distribution. Then define the following scores: 

Ej = E[MU^.j)], (3.4.4) 



or 



E. = Mj/N+1), (3.4.5) 

j = 1 , . . . , N. Define 



and 



1 
(j)* = / (J) (u)du (3.4.6) 





A^ = / (()2(u)du - (<j.*)2, (3.4.7) 

• 



and consider the statistics 



'' U 



V = [ Z (X-;-^)E ]/(AN^a ) (3.4.8) 

j=l ^ ^ij ^ 



i - 1, 2, ..., k, where R . . is the rank of Y.. among Y.,. 

ID 2.J ^ il ' 

^12' •••' ''^iN' ^^® observations of the ith line. Recall 
that statistics such as (3.4.8) are used in the single line 
setting to test hypotheses about the slope. 



144 



Assume F, the underlying error cdf of the assumed model 
(3.1.1), is absolutely continuous and 

CO 

IMF) = / [|^]2dF(x) < <.. (3.4.9) 



Then we again reserve the symbol ¥ (u) for the optimal score 
function: 



T(u) = -^IJlImI , < u < 1. (3.4.10) 

f(F ^(u)) 



It can be shown, 



1 

/ 'y (u) du = , and 




1 2 
/ Y (u)du = I* (F) 





We also define 



P(^,<t>) = [/ 'i'{u)<f (u)du]/[A^I*(F)]^, (3.4 11) 





which can be regarded as a measure of the correlation 
between the chosen score function cp and the optimal one {"H) 



145 



I 



for the error distribution being considered. The expres- 
sions p(T,(j)) (3.4.11) and I* (F) (3.4.9) Vi7ill appear in the 
development of the statistic based on Sen's (1969) work. 

We now define Sen's statistic in the case of several 
regression lines and then give our modification. Let 



V^(y^+bx) (3.4.12) 



denote the value of V. (3.3.7) based on Y . , + bx, , Y.^ + 

1 il 1' i2 

bx^, ..., Y.^ + bx . Define 



V = r V./k. 
i=l 



Assuming H^ , let 3* denote the Hodge s-Lehmann estimate of 
the common slope of all k lines based on V. Define 



^i " ^i(Xi-^*2i) ' (3.4.13) 

i = 1, 2, ..., k. Then Sen proposed the statistic 



k ^2 
L = Z V. (3.4.14) 

i=l ^ 



to test Hq (3.4.1) against H^_^ (3.4.3). The statistic L is 
a quadratic form in the V. , i = 1 , . . . , k. We gave an 



146 



intuitive motivation for the forin of L in Chapter Tvro. We 
now define the modification of L. 

Let 3| denote the Hodges-Lehmann estimate of 6 . based 
on V^ (3.4.8), that is, based only on the observations from 
the ith line. Let 

Vj_ = V. (Y^-B*x)/ (3.4.15) 

i = 1, ..., k, and define 



A k - 

L = Z (V.-V)2, (3.4.16) 

i=l ^ 



- k . 
where V = Z V./k. Since we are interested in rejecting H. 
i=l ^ 

only when the slope of one of the k-1 lines differs from the 
slope of the kth line, we propose transforming the observa- 
tions by using the estimate of the slope of the kth line, 
3*, rather than the pooled slope estimate, 6*. We 
again consider a quadratic form, now in the V., i = 1, ..., 
k. Thus we have made a reasonable modification in the form 
of Sen's original statistic. The basic asymptotic theory 
still holds, and proofs leading to the specific form of a 
test of H_ against H- based on L will be given. 

We now state the basic set of assumptions required for 
the results in this section. These are the assumptions Sen 



"■^-1 TTf^-nrrar-r-. 



147 



gives, applied to the setting of common regression 
constants: 



^ - 2 
A. Z (x.-x) •> <» as N ^- <=°. 

j=l ^ 



max (x.-x) 

B. l^i^ N ^ 

— jj -^ as N ^ 

2 (x.-x) 2 

j=l ^ 



C. F is absolutely continuous with (3.4.9) 

I* (F) < 00, 



D. The score function <f) (u) is an absolutely 
continuous and non-decreasing function of 
u, < u < 1, that is square integrable 
over (0,1) . 

We note that conditions A and B are identical to conditions 
I and II, respectively, defined previously. All four of 
these conditions will be referred to jointly in the follow- 
ing proofs as conditions A-D. Finally, we state again the 
sequence of alternatives we consider: 



n^: 3. = 63^ + co./N^k^a^ (3.4.17) 



-L. X f • * • f }\. f 

where 0)^ = 0. Most of the following lemmas are proved in 
the work of Sen (196 9) . 



■^t-S-,V---W^- .,?Ki^ii 



148 



Lemma 3.4.1 ; Under H^ (3.4.17) and the conditions 



A-D, 



N^a^(B^-3|) I = Op(l) (3.4.18) 



and 



N^0^(6^-S*)| = (1) (3.4.19) 



as N •» «>, i = 1, . . . , k. 
Proof of leiruna: 



The result (3.4.18) is equation (3.20) in Sen (1969, 
p. 1675) , when using the notation of this section. Result 
(3.4.19) follows from (3.4.18): 
Under E^: ^^ = &^ + <^i/N^k^ax, 



|N\(6.-e*) 



= lN^o^^(6k-3p + a)^/k'^ 



- I^^'^x^^k-^k^ 1 + Ui/k^ 



= (1) + |ajj_/k^| (from (3.4.18)) 



= (1) , as N ^ 

XT 



149 



Lenmia 3.4.2 ; Under H (3.4.17) and the conditions 
A-D, 

|V.(Y.-3|X)| = Op(l) (3.4.20) 

as N ^ ", i = 1, . . . , k. 
Proof of lemma ; 

The result (3.4.20) is equation (3.22) in Sen (1969, 
p. 1675) , when using the notation of this section. Taking i 
= k, we note (3.4.20) yields: 



V. 



kl = l\(Ik-^k^H = °p(l) (3.4.21) 



as N ■> " . ■ 

Lemma 3.4.3 ; Let a^ be a positive constant and define 
I(aQ) = {a: |a| 1 a^ } . Then assuming conditions A-D, 

Vi(Ii-[Bi-a/N^a ]x) - V. (Y . - [ g, -b/N^a ]x) 

= p ('l',(j)) (a-b) [I*(F)] + o (1) (3.4.22) 



holds simultaneously for all a, b e I(a_) as N 



Proof of lemma : 

This is proven as Lemma 3.2 in Sen (1969, p. 1674). We 
have restated the lemma here in the notation of this 
section. 



150 



Remark ; Sen notes that since (3.4.22) holds 
simultaneously for all a, b e i (a^), it follows that 
(3.4.22) also holds if a and b are random variables 
depending on N and both are (1) as N ^ «. 

Lemma 3.4.4: Under H^ (3.4.17) and the conditions 
A-D, 



^i " ^i^^i " ^k^^ = P('l',<(') [I*{F)]\^a^(6*-e*) + o (1) 

(3.4.23) 

as N ^ «>, i = 1, . . . , k. 
Proof of lemma: 



Vi = [V.(Y^-B*x) - V.(Y.-6|x)] + Vi(Y.-6^x). (3.4.24) 



Let 



a = N^a^(6.-6*) (3.4.25) 



and 



b = N^c:^(6.-6^). (3.4.26) 

By (3.4.19) and (3.4.18), a = (1) and b = O (1) as N > -. 

P p 

Then in view of the remark following the proof of Lemma 
3.4.3 we may apply that lemma with a and b as in (3.4.25) 
and (3.4.26), respectively, obtaining 



151 



Vi(li-ejx) - v3_(Y^-ejx) 



= P (^F, (I)) [I*(F) ]\^a^(B^-6*) + o (1) (3.4.27) 



as N -> ", i = 1, ..., k. Apply (3.4.27) and (3.4.20) to 
(3.4.24) : 

Vi = [V^(Y.-e*x) - V^(Y.-B|x)] + V. (Y.-6^x) 



= [p(Y,*) [I*(F)]^N'^a^(e^-e*) + o (1)] + o (1) 



as N ^ 00, i = 1, _.^ k^ Thus (3.4.23) is verified. 

Lemma 3.4.5; Under H^ (3.4.17) and conditions A-D, 



^''^x^^i-^k^ f N(kA.,[p2('F,,j,)I*(F)]-l) (3.4.28) 



as N -> =0, i = 1, ..., k. 
Proof of lemma ; 

By Lemma 3.4 in Sen (1969, p. 1676), 



N^cr^(e^-e.) f N(O,[p2(*,0)I*(F)]-^) (3.4.29) 



as N ->- ". Now under H„, 

— N' 






- ^T'2 



'- 1?. 3^ 



N^a^(3*-e^-,./N^k^a^) 



152 



= N^a^(6|-6j^)-(a)^/k^) 



, In view of (3.4.29), this implie: 



%v d 



-1, 



N^a^(B|-Bj^) - (oj^/k^) ; N(0,[p''('^,<j,)I*(F)]"^) 



as N ■* ", and the desired result (3.4.28) follows 
immediately. 

Lemma 3.4.6: Let 



X 



X 



X = 

— n 



2 ,n 



X 



k,n 



X = 
— n 



.o 




•l,n 




o 




2,n 




• 




• 




. 




o 




k-l,n_ 





1 ,n k,n 
2 ,n k,n 



k-l,n k,n 





' 




^1 




^2 

• 


p = 


• 


1 


^k 



o 

u = 



'^l-^k 



^2-^k 



^k-l-^k 



(3.4.30) 



where X^, n >_ 1 represents a sequence of random vectors and 
_M is a vector of constants. Let a > be a constant. If 



Xn f ^k^ii'^'V 



(3.4.31) 



as n ->- <» , then 



'■*»»«^ »«W,— -: JSr -i^- - 



153 



2i° t Nk-i(l°'^^(Vi+Jk_i)) (3.4.32) 



as n -*■ °= . 
Proof of lemma; 



Consider an arbitrary linear combination of the 

elements of X°: 
— n 



k-1 ^ k-1 

rO 



2 ^^X. ^ = Z A. (X. -X, ) 
i=l 1 1'^ i=i 1 i.n k,n' 



k-1 k-1 

= r x.x. ^ - X, z A. . 

^^^ 1 i,n k'^i=i 1 



Since X^ has a multivariate normal limiting distribution it 

follows (Serf ling, 1980, p. 18) that any linear combination 

■ 
of the X^ must converge in distribution to the same linear 

combination of a random variable with that multivariate 

normal distribution. Hence 



^-1 k-1 k-1 k-1 „ k-1 _ 

,i,^i^i,n - ^k,n.i/i . N(.Z^X...-,^_Z_^A.,a2r(_z^.2 



) + 



k-1 ^ 



(.^ ^i) ]) (3.4.33) 



as n ^ °o . Letting 



A = 



k-1 



(3.4,33) can be expressed as follows 



154 



A'^n t NU'y°,a2A'(Ik_i+Jk_i)i) 



(3.4.34) 



as n -> ". The result (3.4.32) follows (Serfling, 1980, 

p. 18), since (3.4.34) implies that any linear combination 

of the elements of X° converges in distribution to the same 

linear combination of a variable with the 
o 2 



N. 



k_l (Ji 'Cr (Ik_i+J]^_2) ) distribution 



Before proceeding with the next lemma, we establish 
some additional notation. Let B^, g* , and co^, i = 1 , . . . , 
k, be as defined earlier and now define 



!* = 



'^ 



^k 



'k 
'k 



'k 

L J 



(3.4.35) 



155 



where ^^ has k elements. For a vector X v/ith k elements, 
let X be the vector of k-1 elem.ents formed by subtracting 
the kth element from each of the first k-1 elements of ^. 
This notation was used implicitly in the statement of Lemma 
3.4.6. For example. 






6* -B* 
■^k-l ^k 



and 03 = 



'"l-'^k 


' 


'^i 


"2-^k 




0)2 


« 




• 


• 




• 


• 




• 


'"k-r^k 




'^k-l 






^ 



(3.4.36) 



since m, = . 



Lemma 3.4.7 ; Let _e*° and u° be as given in (3.4.36), 



and let 



-1 



t^ = [p''('i',<^)I*(F)]"-'(Ij^_^+j^_^). 



(3.4.37) 



Under H (3.4.17) and conditions A-D , 



-N 



N' 



-O. d 



^x<i* ) :Nk-i(^"^^ 'M 



(3.4.38) 



as N -> °° . 
Proof of lemma; 



Recall the definition of _6, and _B* in (3.4.35). From 
Lemma 3.4.5 and the independence of g* and ^"^ , , for i 7^ i ' , 



= »^ i .— ^1-..— — f.^--.. ., ,—— ^-- ^ ^■^- ..„ - - , — - ^ , , r - 



156 



it follows that 



N^^d*-!^) J Nj^{k-^,[p2(Y,^)i*(F)]-lT ) 



as N > CO, under E^. Now we identify N^a^{B*-3^) with X 

X — — K — n 
(3.4.30) and apply Lemma 3.4.6: 



N^x([l*-lk]°) t N(k-V,[P^(^,*)I*(F)](Ij^_^+J^_^)), 



* 



o „*o 



^^k-l)) 



which, since [^ -_3 ] ° = 3 °, implie 



n'^^^CB*") f N{k-V.[P^^.<l>)IMF)](Ij^_^ 



as N ^ =0, and (3.4.38) has been proved. 



« 

For later use we note that (3.4.23) together with the 

i^ if ie 

fact that N'a^(B^-B^) has a normal limiting distribution 
(3.4.38) implies 



^i = °p(l) (3.4.39) 



under H^ and conditions A-D, asN^oo, i = i^ ^ ]^_2 
By Theorem 8.3.4 in Graybill (1969, p. 171), 



V^ = [P^(^,*)IMF)](I^_^-k-lj^_^). (3.4.40) 



157 



Leinma 3.4.8 : Under H (3.4.17) and conditions A-D, 



Na^(l*°)'lv^(l*°) f xJ_i(p2(^,q))I*(F)k-l Z i.,-Z}h 



z 
i=l 

(3.4.41) 



as N ->• ". 
Proof of lemma : 

From lemma 3.4.7 we have the following; 



•'V*°^''ic-i'x"V4v) 



as N -> ". 



~k 



Let X be a random vector with the N,_ (k~''a) , | ) 
distribution. Then it follows (Serfling, 1980, p. 25) that 



Na 

X 



2(B*°)'|;^(1*°) ^ X'tv^X (3.4.42) 



as N -> CO. The distribution of X'ty^X (Serfling, 1980, 
p. 128) is 



2 ,1 -1 o ' x-l Ov 
Xk_l (k u ?v ii^ ) • (3.4.43) 



Evaluating the noncentrality parameter in (3.4.43), we have 



158 



0)° tyV = P^(^,<J>)I*{F) [a)°'lj^_^(.°-k"V'jj,_iiii°] 



i=l ^ i=l ^ 



k 
p^(Y,<t>)I*(F) E (co.-;;;)^, (3.4.44) 

i=l ^ 



which completes the proof of Leirnna 3.4.8. 

We now give the main result of this section, a theorem 
specifying the asymptotic distribution of the proposed test 
statistic L (3.4.16) under H,,. 

Theorem 3.4.9 ; Under H^^ (3.4.17) and conditions A-D, 



L t Xk_i(P^(Y.<!>)IMF)k"\z {i^^-Z)^) (3.4.45) 



k 
as N -> 00, where L is given by (3.4.16) and oj = Z cj./k. 



i=l ^ 



Proof: 



k — I 

L = Z (V.-V)^ i 

i=l ^ 



= [ z vl'] - [k -^( Z V.)^] 
i=l ^ i=l ^ 



159 



i=l ^ ^ i=l ^ i=l ^ k 



= Z V^ - k ( Z V )^ + V^ - 2k~-^V, Z v.. (3.4.46] 
i=l i=l ^ ^ ^i=l 1 



By Lemma 3.4.2 (see (3.4.21)) 

l\I = Op(l) (3.4.47) 

as N > ", and in (3.4.39) we noted 

IV^I = Op(l) (3.4.48) 

as N ^ 06, i = 1^ — ^ ]^-l^ Consequently, applying these 
results to (3.4.46) yields 



k-1 k-1. 
L = Z V - k ( Z V )^ + o (1) (3.4.49) 

i=l ^ i=l ^ P 



as N ^ 00. By Slutzky's Theorem (Randies and Wolfe, 1979, 
p. 424) it follows from (3.4.49) that if L has a limiting 
distribution as N ^ oo, it is the same as the limiting 
distribution of 



k-1 k-1. 
L^ = Z V - k~'( Z V )^ (3.4.50) 

1=1 i=l 



160 



Leirana 3.4.4 showed the following: 

Vj, = p(¥,ct,) [lMF)]\^a^(iB|-B*) + Op(l) (3.4.51) 

asN->", i = l, ...,k. Once again appealing to Slutzky's 
Theorem, it follows from (3.4.51) and the continuity of the 
quadratic form (3.4.50), that the limiting distribution of 
L^ (and hence of L) is the same as the limiting distribution 
of 



2 2 ^"^ 9 1 k-1 o 

p ('J',f)I*(F)Na^[ Z (e:f-6*)^ - k~-^( E (6*-3*))^]. 

^ 1=1 ^ ^ i=l ^ ^ 

(3.4.52) 

Using the matrix notation _3*° (3.4.36) and t^^ (3.4.40), the 
expression (3.4.52) is equal to 



Na;(l*°)'t;^l*°). (3.4.53) 



From Lemma 3.4.8, (3.4.53) has a limiting (N^") 

2 2 -1 ^ ? 

Xj^_l(p {'V,<p)I*{F)k Z (o) -to)'') distribution so (3.4.45) 

i=l 

holds and the proof is complete. 

The asymptotic distribution of L (3.4.16) under 

2 

^0" ^1 = ^2 " • • • " ^k ^^ ^k-l^°^' "^^^^ follows from 

Theorem 3.4.9 if w^ = 03^ = ... = oi^ = . As used in 
(3.2.77), let ^y^_^ ^ be the upper 100 (1-a) percentile of the 



161 



Xj^_-[^(0) distribution. Then the test of H against H 
(3.4.1) that rejects H when 



^ - ^k-l,a (3.4.54) 

is an approximate level a test. Comparisons of the test 
(3.2.77) based on S (3.2.74) with this test will be made in 
the next section by calculating the tests' asymptotic 

relative efficiency. 

3.5 Asymptotic Relative Efficiencies 
In Sections 3.1 and 3 . 2 we proposed and developed a 
test comparing the slopes of several regression lines with a 
standard or control. A competitor to this proposed test was 
suggested in Section 3.4. We now compare these two tests by 
examining the Pitman asymptotic relative efficiency (PARE) 
of the corresponding test statistics. 

Recall in Section 3.2, assuming model (3.1.1), the 
proposed test of the null hypothesis 

-0* ^1 " ^2 " ••• " ^k' (3.5.1) 

against the alternative, 

-1" '^i ^ ^k ^°^ ^^ least one i, (3.5.2) 



162 



was based on 



S - T't ^T, (3.5.3) 



where T is a k-1 dimensional vector with elements 



T^(0) = (Na^^^)"^ i:Zx^gSgn(Z^g-Z^^), (3.5.4) 



i = 1, ..., k-1. The Statistic T.(0) compares the slope of 
the ith line with the slope of the kth line. Statistics 
having the form of T. (0) were proposed and discussed at 
length in Chapter Two, when comparing the slopes of two 
lines. The matrix |~ appearing in (3.5.3) is the inverse 
of the asymptotic covariance matrix of T (see (3.2.73)). 

In the previous section, we introduced a second 
statistic for testing H , based on a statistic proposed by 
Sen (1969). Again assuming model (3.1.1), recall the 
statistics 



V^ = [ Z (x.-x)Ej^ ]/(AN'a ), (3.5.5) 

j = l -■ ij 



i = 1, ..., k, where R . . is the rank of Y. . among the 

observations of the ith line, E_ are the scores (3.4.4) or 

ID 

(3.4.5), and A is the constant in (3.4.7). Let Bf, denote 
the Hodges-Lehmann estimate based on V, and let 






163 



Vi = V.(Y.-3*x) (3.5.6) 

equal the value of V. when using Y. - B^x^ , Y. - B*x 
••" '^iN ~ ^k^N' ^^^^ ^^^ statistic based on Sen's work is 



^ k - 

L = Z (V.-V)^ (3.5.7) 

i=l 



where V = Z V./k. 
i=l ^ 



Let co^, 012, ..., (jjj^ be given constants with u = 0. 
For use in the next result define 



-1 '^ _ p 

^2 _ _i=l_ 

^w =2 ' (3.5.8) 

0) 



-1 ^ 
where w = k E oj . . We derived the asymptotic distribu- 
i=l ^ 



tions of S and L under the sequence of alternatives. 



H^: ^i = ^k ^ a)^/N^k^a^, (3.5.9) 



-L -1-f •••^ jC. 



164 



These asymptotic distributions will be used to arrive at the 
PARE of the two statistics. 

Recall the test (3.2.77) based on the generalization of 
the Sievers-Scholz statistic, S (3.5.3), depends on the 
underlying error distribution F through A* = 12A (F) . In 
practice we suggested a conservative version of the test 
(3.2.77) based on replacing A with A given by (3.2.84). 
However for asymptotic efficiency comparisons investigating 
the relative merits of the Sievers-Scholz and Sen methods, 
it is appropriate to use the test (3.2.77) based on S. The 
resulting PARE expressions will be evaluated assuming 
different specific error distributions. 

Result 3.5.1; Let c^, 1(g), a*, p(^,(j)), and I* (F) be 

as defined in (3.5.8), (3.2.13), (3.2.12), (3.4.11), and 

(3.4.9), respectively. Assuming conditions I-IV and A-D, 

the PARE of S (3.5.3) with respect to L (3.5.7) under H 

— N 

(3.5.9) is 



^2 * 

PARE(S,L) = ^^^ W) [1 + 1 - 2A 

(l-A*)p^(Y,*)I*(F) (l+(k-2)A*)c^ 

0) 



(3.5.10) 



Verification of Result 3.5.1 ; Under H^^ and the assumed 
conditions. Theorems 3.2.8 and 3.4.9 give the limiting 
distributions of S and L, respectively, as follows: 



165 



k-1 ^ k-1 ^ 



c 



(l+(k-2)A*) Z cor-A*{ E 0).) 
(1-A*) (l+(k-2) A*) 



(3.5.11) 



k 
L f X? 1 (p^(^,<j>)I*(F)k~^ E (to.-u)^). (3.5.12) 

^ ^ i=l 1 



VJith some algebraic manipulations, the noncentrality 
parameter for S in (3.5.11) can be expressed as follows; 



2 k * 
121 (g) .,-1 „ . -x2,r. 1 - 2A ,- _ -^, 
* t^ ^ ((i^--w) J [1 + J 5"]. (3.5.13) 

1-A i=l -^ (l+(k-2)A )c 

CO 



Since both S and L have limiting noncentral chi-squared 
distributions under H with equal degrees of freedom, it 
follows, as discussed in the verification of Result 2.4.5, 
that the PARE(S,L) is given by the ratio of the 
noncentrality parameters. Then using the noncentrality 
parameters for S and L given in (3.5.13) and (3.5.12), 
respectively, the PARE (3.5.10) follows imm.ediately . 

The PARE(S,L) (3.5.10) depends on the constants 

2 
,, w-f '•'/ u, through c (3.5.8). Hence we give bounds 

for this PARE that are independent of the to ' s . We first 

state and prove a needed lemma. 



tii 



Lemma 3.5.2 : For arbitrary constants oi^ , 



CJ, 



,^, . . . , ^^ 

satisfying w, = and m . ^ for some i = 1, ..., k-1, 
define 



166 



1 
0). = (1/i) s u and (3.5.14) 



2 ^ - 2 
^i - .^ ((Jj-o)^) (3.5.15) 



for i = k-1, k. Then 



1,-2 
V 
-2~ - ^~^- (3.5.16) 

^k 



Proof of lemma; 



First note 



""k-l = (k/k-l)^;;;^. (3.5.17) 



Then 



= 2 _ ^ 2 , -2 

^ i=l ^ k 



,E^ .2 _ (3,_i,-2_^ ^ (k-l)^2_^ _ ^-2 



^k-1 ^ ^^ /k-l)a)^_i - k[;;J (from (3.5.17)) 



167 



= sj_^ + kmj[(k/(k-l)) - 1] 



^k-1 + [k;;;^/(k-i)] 






, -2 
k 
—2 = (k-l)[l - (sj_^/sj)]. ■ (3.5.18) 

^k 



The right hand side of (3.5.18) is maximized when s"^ =0 

k-1 ' 

in which case (3.5.16) follows immediately. 

Result 3.5.3: The PARE (S,L) given by (3.5.10) has the 
following bounds: 



121 (g) * 1 ot2 / ^ 
■* 2~^ * 1 P^^^ (S,L) < — ^^^ ^?/ 



(1-A )p" (¥,<(,) I'" (F) ■ ■ - A*p2('^,<^)l*(F) 

(3.5.19) 

Verification of Result 3.5.3 : Recall the bounds for 
A = 12A (F) , 



1/3 < A < 1/2, 



applicable under continuous F. Then since 1-2 A* > , it 
follows that 



* 

1 - 2A 

* 2 - ° (3.5.20) 



(l+(k-2)A )c 

0) 



■"a'MB>Btww'»iair3i^— f— — niiuM n 



168 



and so the lower bound in (3.5.19) is obtained by applying 
(3.5.20) to the PARE (3.5.10). To obtain the upper bound, 
first note that (3.5.16) of Lemma 3.5.2 can be expressed as 



— 1 k-1. (3.5.21) 

c 



Applying (3.5.21) to the PARE (3.5.10) yields 



2 

PARE (S,£) < ^-i2I__(|)_ ^ k ^^ ^ (3.5.22) 

p (Y,(j))I (F) l+(k-2)A 



Then since 



A* < 1/2 => ^ ^ -X_, (3.5.23) 

l+(k-2)A A 



the upper bound in (3.5.19) is obtained by applying (3.5.23) 
to (3.5.22). 

Suppose we assume Wilcoxon scores, 4)(u)=u, 0<u<l. 
Then, as derived in Chapter Two, it follows from (2.4.34), 
(2.4.35) , and (2.4.39) that 



P^('i',*)I*(F) = 12l^(f) . (3.5.24) 



Hence we substitute (3.5.24) into (3.5.19) resulting in 



t" 'Ifll.BHI' 'lipr^'ir ■ iiil'l j-llMLli.ili 



169 



i^(g) 



(l-A*)I^(f) 



< PARE (S,L) < 



t2. 



£L 



— * 2 ' 
A I^(f) 



(3.5.25) 



assuming <i> (u) = u. Since A = 12A (F) , values of A may be 
obtained from Table 2 4 for several common error distribu- 
tions. 

Table 25 gives the upper and lower bounds of the 
PARE(S,L) for four error distributions, assuming (p (u) = u. 
We see that the PARE (S,L) is close to one for the three 
distributions with light to moderately heavy tails (uniform, 
normal, double exponential) , but has value close to 1/2 
under the heavily tailed Cauchy distribution. These results 
are essentially the same as the PARE comparisons of the 
Sievers-Scholz and Sen statistics in Table 2 for the two 
line setting. Thus, as remarked at the beginning of this 
chapter, the superiority of Sen's test with respect to 



Table 25. Upper and Lower Bounds of PARE(S,L) for Selected 
Error Distributions Assuming (u) = u. 



Distribution 



Uniform 

Normal 

Double Exponential 

Cauchy 



Lower Bound 



Upper Bound 



0.87 
0.97 
0.74 
0.46 



0.91 
1.04 
0.83 
0.55 



170 



the Sievers-Scholz test in terms of PARE is not surprising, 
given our results in Chapter Two and the fact that Sen's 
test maximizes the efficiency relative to the likelihood 
ratio test. Note, however, that assuming (f) (u) = u, the PARE 
of Sen's test to the classical least squares theory test 
under double exponential errors is 1.50 (Sen, 1969, p. 1676; 
Hollander and Wolfe, 1973, p. 64) and the corresponding 
bounds of the PARE of the Sievers-Scholz test based on S to 
the classical test are 1.11 and 1.24. The point here is 
that although the PARE (S,L) favors Sen's test under heavily 
tailed distributions, the difference between the tests' 
PARES with respect to the least squares theory test is not 
impressive. Also, recall from Section 3.3 that the 

statistic S has an advantage over L since the form of S 

s 
allows the construction of exact conditional ta'ts of H. that 

are computationally feasible under small sample sizes. The 

iterative computations necessary to perform Sen's test 

preclude the possibility of computationally feasible, exact 

tests based on his statistic. Furthermore, simulation 

studies by Lo, Simkin, and Worthley (1978) indicate the 

power of Sen's test with respect to the classical least 

squares test in the case of three regression lines under 

small samples is quite conservative. In view of these 

points, the proposed generalization of the Sievers-Scholz 

method to the setting of several regression lines has merit 

as a robust, computationally simple technique allowing exact 

tests that are feasible under small sample sizes. 



g*TT * —t tm-- i 



CHAPTER FOUR 
CONCLUSIONS 

We now summarize the conclusions resulting from the 
work in Chapter Two and Chapter Three. In Chapter Two we 
considered the case of two regression lines. Assuming 
common regression constants for the two lines, we proposed 
the application of a statistic due to Sievers and Scholz 
(2.1.2) to the observed differences Z., j=l, ..., n, given 
by (2.1.3). The proposed statistic (2.1.2) employs weights, 
^rs' ^^^^ ^^® chosen by the user'. If the regression con- 
stants are x^, , x^^, then when using the weights 

^^2 ~ ^s ~ ^r' ^^^ null distribution of the proposed statis- 
tic depends on the regression constants. An associated 
exact confidence interval for the slope difference can be 
obtained by calculation of a permutation distribution. We 
found this to be feasible with the use of a computer when 
the number of regression constants was small, say less than 

eight per line. When using the weights a = 1 if x > x . 

rs s r' 

zero otherwise, the proposed test statistic essentially 
reduces to a statistic due to Theil and Sen (2.1.4), com- 
puted using the observed differences Z . , j = 1 , . . . , n. In 
this case, the null distribution of the proposed statistic 
depends on Kendall's tau. An associated exact confidence 

171 



172 



interval for the slope difference can be calculated using 

readily available tabled critical values of this distribu- 
tion. VJe proposed this procedure when use of the weights 

a = X - X was not feasible, 
rs s r 

Pitman asymptotic relative efficiencies (PAREs) of 

these two proposed tests with respect to the tests of 

Hollander, Rao and Gore, Sen, and the classical t-test were 

computed assuming equal spacing of the regression constants. 

Choice of the weights a = x - x or the zero-one weights 

^ rs s r ^ 

does not affect the PARE under equal spacing, so results are 
given jointly for these two methods. The PARE of the 
proposed method relative to Hollander's method is greater 
than one over a wide variety of error distributions. The 
PARE of the proposed method with respect to the Rao-Gore 
technique is 4/3 irrespective of the underlying error 
distribution (subject to certain regularity conditions) . 
Simulation results assuming unequal spacing of the regres- 
sion constants also favor the proposed methods over the 
other two exact procedures (the Hollander and Rao-Gore 
methods) . 

The PARE of Sen's (1969) test with respect to the 
proposed test is greater than or equal to one under most 
common error distributions. However, Sen's approach is only 
asymptotically distribution-free and requires iterative 
techniques. The PAREs and simulation results showed the 
classical t-test performs better than the proposed methods 
under distributions with light tails. However, the 






173 



classical test does very poorly when the underlying error 
distribution has heavy tails, such as the Cauchy distribu- 
tion. For these reasons we feel the proposed methods are 
preferred when exact, distribution-free procedures are 
desired to make inference about the slope difference, 
assuming common regression constants are used for both 
lines. 

In Chapter Three we considered the case of several 
regression lines. Assuming common regression constants, we 
defined a set of statistics, each having the form of the 
proposed statistic of Chapter Two. Specifically, when 
comparing the slopes of k lines, k-1 statistics were con- 
sidered. The ith statistic compared the slope of the ith 
and kth lines. We proposed a test based on a quadratic form 
(3.2.74) in this set of k-1 statistics. Hence this test was 
designed to detect alternatives where one or more of the 
slopes of the first k-1 lines differ from the kth line. 
This is the case where the kth line is considered to be a 
standard, or control. 

A modification of Sen's (1969) test was constructed as 
a potential competitor to the proposed test. Loxver and 
upper bounds of the PAREs of the proposed test with respect 
to the modification of Sen's test were calculated for 
selected error distributions (see Table 25) . Although these 
bounds favor Sen's test under the very heavily tailed Cauchy 
distribution, they are close to one for the uniform, normal, 
and double exponential distributions. We showed that exact 



174 



conditional tests based on the proposed statistic are 
computationally feasible under small sample sizes, a feature 
not shared by Sen's statistic. Hence the method proposed in 
the setting of several regression lines has merit as a 
relatively robust, computationally simple technique allowing 
exact tests. 



BIBLIOGRAPHY 

Adichie, J.N. (1967) . Estimates of Regression Parameters 
Based on Rank Tests. Ann. Math" Stat. , 38, 894-904. 

Adichie, J.N. (1974). Rank Score Comparison of Several 
Regression Parameters. Ann, of Stat. , 2, 396-402. 

Boyett, J.M., and Shuster, J.J. (1977). Nonparametric 

One-sided Tests in Multivariate Analysis with Medical 
Applications. J. Amer. Stat. Assoc , 72, 665-668. 

Chow, Y.S., and Teicher, H. (1978). Probability Theory; 
Independence, Interchangeability , Martingales . New 
York: Springer-Verlag. 

Crow, E.L., and Siddigui, M.M. (1967). Robust Estimation 
of Location. J. Amer. Stat. Assoc , 62, 353-389. 

Dwass, M. (1957). Modified Randomization Tests for Non- 
parametric Hypotheses. Ann. Math. Stat. , 28, 181-187. 

Finney, D.J. (1964) . Statistical Method in Biological 
Assay . London: Charles Griffin and Company. 

Graybill, F.A. (1969) . Introduction to Matrices with 
Applications in Statistics . Belmont, California: 
Wadsworth. 

Hajek, J. (1962) . Asymptotically Most Powerful Rank Order 
Tests. Ann. Math. Stat. , 33, 1124-1147. 

Hajek, J., and Sidak, Z.S. (1967). Theory of Rank Tests . 
New York: Academic Press. 

Hodges, J.L., and Lehmann, E.L. (1963). Estimates of 

Location Based on Rank Tests. Ann. Math. Stat., 34, 
598-611. 

Hoeffding, W. (1951) . Optimum Nonparametric Tests. Proc 
Second Berkely Symp. Math. Stat. Prob. , 1, 82-92. 

Hollander, M. (1970) . A Distribution-Free Test for 

Parallelism. J. Amer. Stat. Assoc , 65, 387-394. 



175 



«^f.*r'».-Tu;:=: 



176 



Hollander, M. , and Wolfe, D.A. (1973). Nonparametric 

Statistical Methods . Nev/ York: John Wiley and Sons. 

Ireson, M.J. (1983) . Nonparametric Regression in the 
Analysis of Survival Data. Ph.D. Dissertation in 
Statistics, University of Florida. 

Jaeckel, L.A. (1972) . Estimating Regression Coefficients 
by Minimizing the Dispersion of the Residuals. Ann. 
Math. Stat. , 43, 1449-1458. 

Jureckova, J. (1971) . Nonparametric Estimate of Regression 
Coefficients. Ann. Math. Stat. , 42, 1328-1338. 

Kendall, M.G. (1955) . Rank Correlation Methods . London: 
Charles Griffin and Company. 

Knuth, D.E. (1973) . The Art of Computer Programming . 
Reading, Mass.: Addison-Wesley . 

Lehmann, E.L. (1963) . Nonparametric Confidence Intervals 
for a Shift Parameter. Ann. Math . Stat., 35, 1507- 
1512. 

Lo, L.C., Simkin, M.G., and Worthley, R.G. (1978). A 
Small-Sample Comparison of Rank Score Tests for 
Parallelism of Several Regression Lines. J. Amer. 
Stat. Assoc. , 73, 666-669. 

Mann, B.L., and Pirie, W.R. (1982). Tighter Bounds and 

Simplified Estimation for Moments of Some Rank Statis- 
tics. Commun. Stat. — Theory an d Methods, 11, 1107- 
1117. "" 

Mehta, C.R., and Patel, N.R. (1983). A Network Algorithm 

for Performing Fisher's Exact Test in r x c Contingency 
Tables. J. Amer. Stat. Assoc , 78, 427-434. 

Noether, G.E. (1949). On a Theorem bv Wald and Wolfowitz. 
Ann. Math. Stat. , 20, 455-458. 

Noether, G.E. (1955) . On a Theorem of Pitman. Ann. Math. 
Stat. , 26, 64-68. 

Pagano, M. , and Tritchler, D. (1983). On Obtaining 

Permutation Distributions in Polynomial Time. J. Amer. 
Stat. Assoc. , 78, 435-440. 

Randies, R.H., and Wolfe, D.A. (1979). Introduction to the 
Theory of Nonparametric Statistics . New York: John 
Wiley and Sons. 



177 



Rao, K.S.M., and Gore, A. P. (1981). Distribution-Free 
Tests for Parallelism and Concurrence in Tv/o-Sample 
Regression Problem. J. Stat. Plannina and Inference, 
5, 281-28 6. 

Scholz, F.W. (1977). Weighted Median Regression Estimates, 
Inst. Math. Stat. Bulletin , 6, 44. 

Sen, P.K. (1966) . On a Distribution-Free Method of Esti- 
mating Asymptotic Efficiency of a Class of Nonpara- 
metric Tests. Ann. Math. Stat. , 37, 1759-1770. 

Sen, P.K. (1968) . Estimates of the Regression Coefficient 
Based on Kendall's Tau. J. Amer. Stat. Assoc, 63, 
1379-1389. 

Sen, P.K. (1969) . On a Class of Rank Order Tests for the 
Parallelism of Several Regression Lines. Ann. Math. 
Stat. , 40, 1668-1683. 

Serfling, R.J. (1980) . Approximation Theorems of Mathe- 
matical Statistics . New York: John Wiley and Sons. 

Sievers, G.L. (1978). Weighted Rank Statistics for Simple 
Linear Regression. J. Amer. Stat. Assoc , 73, 628-631. 

Smit, C.F. (1979) . An Empirical Comparison of Several 
Tests for Parallelism of Regression Lines. Commun . 
Stat. — Simula. Computa. , 8, 61-74. 

Terry, M.E. (1952) . Some Rank Order Tests Which are Most 
Powerful Against Specific Parametric Alternatives. 
Ann. Math. Stat. , 23, 346-366. 

Theil, H. (1950) . A Rank-Invariant Method of Linear and 
Polynominal Regression Analysis. I, II, III, 
Koninklijke Nederlandse Akademie van Wetenschappen, 
Proceedings , 53, 386-392, 521-525, 1397-1412. 



BIOGRAPHICAL SKETCH 

Raymond Richard Daley was born in West Palm Beach, 
Florida, on March 3, 1957. He moved to Titusville, Florida, 
in 1962. After graduating from Titusville High School in 
1975, he enrolled at the University of Central Florida. He 
received his Bachelor of Science degree in statistics in 
1978 and was honored at his commencement for having attained 
the highest grade point average of his graduating class. He 
entered Graduate School at the University of Florida in 1978 
and later received his Master of Statistics degree in 1980. 
He expects to receive the degree of Doctor of Philosophy in 
December, 1984. 

His professional career has included work as a statis- 
tician for an Air Force climatology study at the University 
of Central Florida and consulting in the Biostatistics Unit 
of the J. Hillis Miller Health Center at the University of 
Florida. He has received Graduate School fellowships and 
graduate assistantships during his academic career at the 
University of Florida. He is a member of the American 
Statistical Association and the Biometric Society. 



178 



_ _I certify that I have read this study and that in my 
opmxon It conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophv. 



\ 



Pen aver V. Rao, Chairman 
Professor of Statistics 



I certify that I have read this study and that in my 
opinion It conforms to acceptable standards of scholarly 
presentation and is fully adequate in scope and quality, as 
a dissertation for the degree of Doctor of Philosophy. 






U M d?a. 



Ronald H. Randies 
Professor of Statistics 



I certify that I have read this study 
opinion it conforms to acceptable standa 
presentation and is fully adequate in sc 
a dissertation for the degree of Doctor o 



and that in my 
s of scholarly 
and quality, as 
f Philosophy. 



rd 

ope 



C^ . CT^ ^ 



JohnjGp Saw 

Professor of Statistics